Resolved: "first derivative of the given function y=cot5x"

Answered question

2022-01-10

first derivative of the given function

$y = c o t 5 x$

Answer & Explanation

nick1337

Expert2022-02-08Added 777 answers

Find the derivative of the following via implicit differentiation:
$\frac{d}{d x} (y) = \frac{d}{d x} (\cot (5 x))$

Using the chain rule, $\frac{d}{d x} (y) = \frac{d y (u)}{d u} \frac{d u}{d x}$ , where $u = x$ and $\frac{d}{d u} (y (u)) = y^{'} (u) :$
$(\frac{d}{d x} (x)) y^{'} (x) = \frac{d}{d x} (\cot (5 x))$

The derivative of $x$ is $1$ :
$1 y^{'} (x) = \frac{d}{d x} (\cot (5 x))$
INTERMEDIATE STEPS:
Possible derivation:
$\frac{d}{d x} (x)$
The limit definition of the derivative for a function $f (x)$ is $\frac{d}{d x} (f (x)) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$ . Let $f (x) = x$ . Then:
$f (x + h) = (x + h)$
Substitute $f (x + h)$ and $f (x)$ into the limit definition for the derivative:
$\frac{d}{d x} (x) = lim_{h \to 0} \frac{(x + h) - x}{h}$
Combine like terms in the numerator:
$= lim_{h \to 0} \frac{h}{h}$
The limit variable h approaches, but is never equal to, $0$ and therefore can be algebraically canceled:
$= lim_{h \to 0} 1$
The limit of any constant is that constant, so simplify to finish computing the derivative:
$\frac{d}{d x} (x) = 1$

Using the chain rule, $\frac{d}{d x} (\cot (5 x)) = \frac{d \cot (u)}{d u} \frac{d u}{d x}$ , where $u = 5 x$ and $\frac{d}{d u} (\cot (u)) = - \csc^{2} (u)$ :
$y^{'} (x) = - \csc^{2} (5 x) (\frac{d}{d x} (5 x))$
INTERMEDIATE STEPS:
Possible derivation:
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All rights reserved{"serverSide":false,"href":"https:\/\/plainmath.org\/differential-calculus\/49794-first-derivative-of-the-given-function","location":"\/differential-calculus\/49794-first-derivative-of-the-given-function","scheme":"https","host":"plainmath.org","port":443,"base":"","pathname":"\/differential-calculus\/49794-first-derivative-of-the-given-function","search":null,"sentry_dns":"https:\/\/d4f0b1ed83bf47f0b61da5ebded3a536@sentry.prokit.me\/77","env":"prod"}{"question":{"id":49794,"h1":"first derivative of the given function","content":"<p>first derivative of the given function<\/p>\n\n<p><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\" space=\"4\"><mjx-c class=\"mjx-c1D450 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45C TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D461 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c35\"><\/mjx-c><\/mjx-mn><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><mo>=<\/mo><mi>c<\/mi><mi>o<\/mi><mi>t<\/mi><mn>5<\/mn><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><\/p>\n","category":{"id":81,"name":"Differential Calculus","title":"Master Differential Calculus Problems with Our Experts","url":"\/post-secondary\/calculus-and-analysis\/differential-calculus","alias":"differential-calculus","seoText":"<div>One of the most famous and challenging aspects of learning calculus and analysis is dealing with differential calculus equations that are not too complex per se if you have an example of differential calculus sample problems that contain both answers and questions. 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continuity","url":"https:\/\/plainmath.org\/post-secondary\/calculus-and-analysis\/differential-calculus\/limits-and-continuity"}],"answers":[{"id":59117,"content":"<p>Find the derivative of the following via implicit differentiation:<br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-mfrac space=\"4\"><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c63\"><\/mjx-c><mjx-c class=\"mjx-c6F\"><\/mjx-c><mjx-c class=\"mjx-c74\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c35\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mi>d<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>=<\/mo><mfrac><mi>d<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mi>cot<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>5<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><\/p><p>Using the chain rule, <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-mfrac space=\"4\"><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D462 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D462 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D462 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mi>d<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>=<\/mo><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>u<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>u<\/mi><\/mrow><\/mrow><\/mfrac><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>u<\/mi><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><\/mfrac><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>, where <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D462 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>u<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> and <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D462 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D462 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"4\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-mo class=\"mjx-var\" size=\"s\"><mjx-c class=\"mjx-c2032\"><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D462 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3A\"><\/mjx-c><\/mjx-mo><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mi>d<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>u<\/mi><\/mrow><\/mrow><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>u<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>=<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msup><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>u<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>:<\/mo><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-s3\"><mjx-c class=\"mjx-c28 TEX-S3\"><\/mjx-c><\/mjx-mo><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-s3\"><mjx-c class=\"mjx-c29 TEX-S3\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-mo class=\"mjx-var\" size=\"s\"><mjx-c class=\"mjx-c2032\"><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-mfrac space=\"4\"><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c63\"><\/mjx-c><mjx-c class=\"mjx-c6F\"><\/mjx-c><mjx-c class=\"mjx-c74\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c35\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mfrac><mi>d<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><msup><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msup><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>=<\/mo><mfrac><mi>d<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mi>cot<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>5<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><\/p><p>The derivative of <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> is <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>:<br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-mo class=\"mjx-var\" size=\"s\"><mjx-c class=\"mjx-c2032\"><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-mfrac space=\"4\"><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c63\"><\/mjx-c><mjx-c class=\"mjx-c6F\"><\/mjx-c><mjx-c class=\"mjx-c74\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c35\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><msup><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msup><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>=<\/mo><mfrac><mi>d<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mi>cot<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>5<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><br>INTERMEDIATE STEPS:<br>Possible derivation:<br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mi>d<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><br>The limit definition of the derivative for a function <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> is <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-munder space=\"4\"><mjx-row><mjx-base style=\"padding-left: 0.039em\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c6C\"><\/mjx-c><mjx-c class=\"mjx-c69\"><\/mjx-c><mjx-c class=\"mjx-c6D\"><\/mjx-c><\/mjx-mo><\/mjx-base><\/mjx-row><mjx-row><mjx-under style=\"padding-top: 0.167em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c210E TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2192\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-under><\/mjx-row><\/mjx-munder><mjx-mfrac space=\"2\"><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c210E TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\" space=\"3\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c210E TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mi>d<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>=<\/mo><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">lim<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>h<\/mi><\/mrow><mo accent=\"false\" stretchy=\"false\">\u2192<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mrow><\/mrow><\/munder><mfrac><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>h<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><mo>\u2212<\/mo><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>h<\/mi><\/mrow><\/mfrac><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>. Let <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\" space=\"4\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>. Then:<br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\" space=\"3\"><mjx-c class=\"mjx-c210E TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\" space=\"3\"><mjx-c class=\"mjx-c210E TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mi>h<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mi>h<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><br>Substitute <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\" space=\"3\"><mjx-c class=\"mjx-c210E TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mi>h<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> and <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> into the limit definition for the derivative:<br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-munder space=\"4\"><mjx-row><mjx-base style=\"padding-left: 0.039em\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c6C\"><\/mjx-c><mjx-c class=\"mjx-c69\"><\/mjx-c><mjx-c class=\"mjx-c6D\"><\/mjx-c><\/mjx-mo><\/mjx-base><\/mjx-row><mjx-row><mjx-under style=\"padding-top: 0.167em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c210E TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2192\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-under><\/mjx-row><\/mjx-munder><mjx-mfrac space=\"2\"><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c210E TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c210E TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mi>d<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>=<\/mo><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">lim<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>h<\/mi><\/mrow><mo accent=\"false\" stretchy=\"false\">\u2192<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mrow><\/mrow><\/munder><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>h<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>h<\/mi><\/mrow><\/mfrac><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><br>Combine like terms in the numerator:<br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-munder space=\"4\"><mjx-row><mjx-base style=\"padding-left: 0.039em\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c6C\"><\/mjx-c><mjx-c class=\"mjx-c69\"><\/mjx-c><mjx-c class=\"mjx-c6D\"><\/mjx-c><\/mjx-mo><\/mjx-base><\/mjx-row><mjx-row><mjx-under style=\"padding-top: 0.167em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c210E TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2192\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-under><\/mjx-row><\/mjx-munder><mjx-mfrac space=\"2\"><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c210E TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c210E TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mo>=<\/mo><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">lim<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>h<\/mi><\/mrow><mo accent=\"false\" stretchy=\"false\">\u2192<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mrow><\/mrow><\/munder><mfrac><mi>h<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>h<\/mi><\/mrow><\/mfrac><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><br>The limit variable h approaches, but is never equal to, <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> and therefore can be algebraically canceled:<br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-munder space=\"4\"><mjx-row><mjx-base style=\"padding-left: 0.039em\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c6C\"><\/mjx-c><mjx-c class=\"mjx-c69\"><\/mjx-c><mjx-c class=\"mjx-c6D\"><\/mjx-c><\/mjx-mo><\/mjx-base><\/mjx-row><mjx-row><mjx-under style=\"padding-top: 0.167em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c210E TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2192\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c30\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-under><\/mjx-row><\/mjx-munder><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mo>=<\/mo><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">lim<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>h<\/mi><\/mrow><mo accent=\"false\" stretchy=\"false\">\u2192<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mrow><\/mrow><\/munder><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><br>The limit of any constant is that constant, so simplify to finish computing the derivative:<br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mi>d<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><\/p><p>Using the chain rule, <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c63\"><\/mjx-c><mjx-c class=\"mjx-c6F\"><\/mjx-c><mjx-c class=\"mjx-c74\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c35\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-mfrac space=\"4\"><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mi class=\"mjx-n\" space=\"2\"><mjx-c class=\"mjx-c63\"><\/mjx-c><mjx-c class=\"mjx-c6F\"><\/mjx-c><mjx-c class=\"mjx-c74\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D462 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D462 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D462 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mi>d<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mi>cot<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>5<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>=<\/mo><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mi>cot<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>u<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>u<\/mi><\/mrow><\/mrow><\/mfrac><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>u<\/mi><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><\/mfrac><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>, where <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D462 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c35\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>u<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>5<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> and <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D462 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c63\"><\/mjx-c><mjx-c class=\"mjx-c6F\"><\/mjx-c><mjx-c class=\"mjx-c74\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D462 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c63\"><\/mjx-c><mjx-c class=\"mjx-c73\"><\/mjx-c><mjx-c class=\"mjx-c63\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D462 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mi>d<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>u<\/mi><\/mrow><\/mrow><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mi>cot<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>u<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>=<\/mo><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><msup><mrow data-mjx-texclass=\"ORD\"><mi>csc<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>u<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>:<br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-mo class=\"mjx-var\" size=\"s\"><mjx-c class=\"mjx-c2032\"><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c63\"><\/mjx-c><mjx-c class=\"mjx-c73\"><\/mjx-c><mjx-c class=\"mjx-c63\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c35\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-s3\"><mjx-c class=\"mjx-c28 TEX-S3\"><\/mjx-c><\/mjx-mo><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c35\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-s3\"><mjx-c class=\"mjx-c29 TEX-S3\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><msup><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msup><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>=<\/mo><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><msup><mrow data-mjx-texclass=\"ORD\"><mi>csc<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>5<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mfrac><mi>d<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>5<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><br>INTERMEDIATE STEPS:<br>Possible derivation:<br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c63\"><\/mjx-c><mjx-c class=\"mjx-c6F\"><\/mjx-c><mjx-c class=\"mjx-c74\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mi>d<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mi>cot<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo d","createdAt":"2022-02-08","user":{"id":10063,"firstName":"nick1337","lastName":null,"fullName":"nick1337 ","shortName":"nick1337 ","email":"rex0945@gmail.com","questionsCount":0,"answersCount":777,"avatar":{"id":6,"title":"Male 2","src":"\/build\/images\/user-picture\/user-6.png","src2x":"\/build\/images\/user-picture\/user-6@2x.png"},"level":"Expert","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"upVotes":0,"questionId":49794,"question":null,"isApproved":true,"isDeleted":false}],"newQuestions":[{"id":104385,"h1":"What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives of second and higher orders? Give examples.","content":"<p>What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives of second and higher orders?<\/p>\n","category":{"id":81,"name":"Differential Calculus","title":"Master Differential Calculus Problems with Our Experts","url":"\/post-secondary\/calculus-and-analysis\/differential-calculus","alias":"differential-calculus","seoText":"<div>One of the most famous and challenging aspects of learning calculus and analysis is dealing with differential calculus equations that are not too complex per se if you have an example of differential calculus sample problems that contain both answers and questions. It is recommended to use your logic and see how differential calculus problems are formed with the help of several equations where you must be <a href=\"https:\/\/plainmath.org\/post-secondary\/calculus-and-analysis\/integral-calculus\/decimals\">solving equations with decimals<\/a>, functions and their derivatives. These are used to calculate the flow of electricity, model some motion, or set the energy flow of an object in a graphical form.<\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2023-03-30","isAnswered":true,"url":"\/differential-calculus\/104385-what-is-the-mixed-derivative-t","absoluteUrl":"https:\/\/plainmath.org\/differential-calculus\/104385-what-is-the-mixed-derivative-t","user":{"id":98338,"firstName":"Pieszkowo3gc4","lastName":"","fullName":"Pieszkowo3gc4 ","shortName":"Pieszkowo3gc4 ","email":"pieszkowo3gc4_@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":2,"title":"Male 1","src":"\/build\/images\/user-picture\/user-2.png","src2x":"\/build\/images\/user-picture\/user-2@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":"","isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":104221,"h1":"How do I find the y-intercept of a parabola?","content":"How do I find the y-intercept of a parabola?\n","category":{"id":81,"name":"Differential Calculus","title":"Master Differential Calculus Problems with Our Experts","url":"\/post-secondary\/calculus-and-analysis\/differential-calculus","alias":"differential-calculus","seoText":"<div>One of the most famous and challenging aspects of learning calculus and analysis is dealing with differential calculus equations that are not too complex per se if you have an example of differential calculus sample problems that contain both answers and questions. It is recommended to use your logic and see how differential calculus problems are formed with the help of several equations where you must be <a href=\"https:\/\/plainmath.org\/post-secondary\/calculus-and-analysis\/integral-calculus\/decimals\">solving equations with decimals<\/a>, functions and their derivatives. These are used to calculate the flow of electricity, model some motion, or set the energy flow of an object in a graphical form.<\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2023-03-26","isAnswered":true,"url":"\/differential-calculus\/104221-how-do-i-find-the-y-intercept","absoluteUrl":"https:\/\/plainmath.org\/differential-calculus\/104221-how-do-i-find-the-y-intercept","user":{"id":98399,"firstName":"Kira Yoder","lastName":"","fullName":"Kira Yoder ","shortName":"Kira Yoder ","email":"kira_yoder_@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":5,"title":"Female 4","src":"\/build\/images\/user-picture\/user-5.png","src2x":"\/build\/images\/user-picture\/user-5@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":"","isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":104056,"h1":"What are the vertices of 9x^2+16y^2=144?","content":"What are the vertices of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"> <mstyle displaystyle=\"true\"> <mn>9<\/mn> <msup> <mi>x<\/mi> <mn>2<\/mn> <\/msup> <mo>+<\/mo> <mn>16<\/mn> <msup> <mi>y<\/mi> <mn>2<\/mn> <\/msup> <mo>=<\/mo> <mn>144<\/mn> <\/mstyle><\/math>?\n","category":{"id":81,"name":"Differential Calculus","title":"Master Differential Calculus Problems with Our Experts","url":"\/post-secondary\/calculus-and-analysis\/differential-calculus","alias":"differential-calculus","seoText":"<div>One of the most famous and challenging aspects of learning calculus and analysis is dealing with differential calculus equations that are not too complex per se if you have an example of differential calculus sample problems that contain both answers and questions. It is recommended to use your logic and see how differential calculus problems are formed with the help of several equations where you must be <a href=\"https:\/\/plainmath.org\/post-secondary\/calculus-and-analysis\/integral-calculus\/decimals\">solving equations with decimals<\/a>, functions and their derivatives. These are used to calculate the flow of electricity, model some motion, or set the energy flow of an object in a graphical form.<\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2023-03-25","isAnswered":true,"url":"\/differential-calculus\/104056-what-are-the-vertices-of-9x-2","absoluteUrl":"https:\/\/plainmath.org\/differential-calculus\/104056-what-are-the-vertices-of-9x-2","user":{"id":98459,"firstName":"hor1paeeja","lastName":"","fullName":"hor1paeeja ","shortName":"hor1paeeja ","email":"hor1paeeja_@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":6,"title":"Male 2","src":"\/build\/images\/user-picture\/user-6.png","src2x":"\/build\/images\/user-picture\/user-6@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":"","isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":104207,"h1":"How to determine the rate of change of a function?","content":"How to determine the rate of change of a function?\n","category":{"id":81,"name":"Differential Calculus","title":"Master Differential Calculus Problems with Our Experts","url":"\/post-secondary\/calculus-and-analysis\/differential-calculus","alias":"differential-calculus","seoText":"<div>One of the most famous and challenging aspects of learning calculus and analysis is dealing with differential calculus equations that are not too complex per se if you have an example of differential calculus sample problems that contain both answers and questions. 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Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2023-03-25","isAnswered":true,"url":"\/differential-calculus\/104207-how-to-determine-the-rate-of-c","absoluteUrl":"https:\/\/plainmath.org\/differential-calculus\/104207-how-to-determine-the-rate-of-c","user":{"id":98475,"firstName":"Landyn Allison","lastName":"","fullName":"Landyn Allison ","shortName":"Landyn Allison ","email":"landyn_allison_@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":3,"title":"Female 2","src":"\/build\/images\/user-picture\/user-3.png","src2x":"\/build\/images\/user-picture\/user-3@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":"","isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":104118,"h1":"Why are the tangents for 90 and 270 degrees undefined?","content":"Why are the tangents for 90 and 270 degrees undefined?\n","category":{"id":81,"name":"Differential Calculus","title":"Master Differential Calculus Problems with Our Experts","url":"\/post-secondary\/calculus-and-analysis\/differential-calculus","alias":"differential-calculus","seoText":"<div>One of the most famous and challenging aspects of learning calculus and analysis is dealing with differential calculus equations that are not too complex per se if you have an example of differential calculus sample problems that contain both answers and questions. 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Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2023-03-23","isAnswered":true,"url":"\/differential-calculus\/104118-why-are-the-tangents-for-90-an","absoluteUrl":"https:\/\/plainmath.org\/differential-calculus\/104118-why-are-the-tangents-for-90-an","user":{"id":98342,"firstName":"Jefferson Simmons","lastName":"","fullName":"Jefferson Simmons ","shortName":"Jefferson Simmons ","email":"jefferson_simmons_@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":7,"title":"Male 3","src":"\/build\/images\/user-picture\/user-7.png","src2x":"\/build\/images\/user-picture\/user-7@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":"","isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":104059,"h1":"How to find the center and radius of the circle x^2+y^2-6x+8y=0?","content":"How to find the center and radius of the circle <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"> <mstyle displaystyle=\"true\"> <msup> <mi>x<\/mi> <mn>2<\/mn> <\/msup> <mo>+<\/mo> <msup> <mi>y<\/mi> <mn>2<\/mn> <\/msup> <mo>-<\/mo> <mn>6<\/mn> <mi>x<\/mi> <mo>+<\/mo> <mn>8<\/mn> <mi>y<\/mi> <mo>=<\/mo> <mn>0<\/mn> <\/mstyle><\/math>?\n","category":{"id":81,"name":"Differential Calculus","title":"Master Differential Calculus Problems with Our Experts","url":"\/post-secondary\/calculus-and-analysis\/differential-calculus","alias":"differential-calculus","seoText":"<div>One of the most famous and challenging aspects of learning calculus and analysis is dealing with differential calculus equations that are not too complex per se if you have an example of differential calculus sample problems that contain both answers and questions. It is recommended to use your logic and see how differential calculus problems are formed with the help of several equations where you must be <a href=\"https:\/\/plainmath.org\/post-secondary\/calculus-and-analysis\/integral-calculus\/decimals\">solving equations with decimals<\/a>, functions and their derivatives. These are used to calculate the flow of electricity, model some motion, or set the energy flow of an object in a graphical form.<\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. 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It is recommended to use your logic and see how differential calculus problems are formed with the help of several equations where you must be <a href=\"https:\/\/plainmath.org\/post-secondary\/calculus-and-analysis\/integral-calculus\/decimals\">solving equations with decimals<\/a>, functions and their derivatives. These are used to calculate the flow of electricity, model some motion, or set the energy flow of an object in a graphical form.<\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2023-03-22","isAnswered":true,"url":"\/differential-calculus\/103771-what-is-multiplicative-inverse","absoluteUrl":"https:\/\/plainmath.org\/differential-calculus\/103771-what-is-multiplicative-inverse","user":{"id":98344,"firstName":"Lilia Mccarthy","lastName":"","fullName":"Lilia Mccarthy ","shortName":"Lilia Mccarthy ","email":"lilia_mccarthy_@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":3,"title":"Female 2","src":"\/build\/images\/user-picture\/user-3.png","src2x":"\/build\/images\/user-picture\/user-3@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":"","isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":103700,"h1":"How to find the continuity of a function on a closed interval?","content":"How to find the continuity of a function on a closed interval?\n","category":{"id":81,"name":"Differential Calculus","title":"Master Differential Calculus Problems with Our Experts","url":"\/post-secondary\/calculus-and-analysis\/differential-calculus","alias":"differential-calculus","seoText":"<div>One of the most famous and challenging aspects of learning calculus and analysis is dealing with differential calculus equations that are not too complex per se if you have an example of differential calculus sample problems that contain both answers and questions. It is recommended to use your logic and see how differential calculus problems are formed with the help of several equations where you must be <a href=\"https:\/\/plainmath.org\/post-secondary\/calculus-and-analysis\/integral-calculus\/decimals\">solving equations with decimals<\/a>, functions and their derivatives. These are used to calculate the flow of electricity, model some motion, or set the energy flow of an object in a graphical form.<\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2023-03-18","isAnswered":true,"url":"\/differential-calculus\/103700-how-to-find-the-continuity-of","absoluteUrl":"https:\/\/plainmath.org\/differential-calculus\/103700-how-to-find-the-continuity-of","user":{"id":98412,"firstName":"Umattinog768","lastName":"","fullName":"Umattinog768 ","shortName":"Umattinog768 ","email":"umattinog768_@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":2,"title":"Male 1","src":"\/build\/images\/user-picture\/user-2.png","src2x":"\/build\/images\/user-picture\/user-2@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":"","isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":103615,"h1":"How do I find the tangent line of a function?","content":"How do I find the tangent line of a function?\n","category":{"id":81,"name":"Differential Calculus","title":"Master Differential Calculus Problems with Our Experts","url":"\/post-secondary\/calculus-and-analysis\/differential-calculus","alias":"differential-calculus","seoText":"<div>One of the most famous and challenging aspects of learning calculus and analysis is dealing with differential calculus equations that are not too complex per se if you have an example of differential calculus sample problems that contain both answers and questions. It is recommended to use your logic and see how differential calculus problems are formed with the help of several equations where you must be <a href=\"https:\/\/plainmath.org\/post-secondary\/calculus-and-analysis\/integral-calculus\/decimals\">solving equations with decimals<\/a>, functions and their derivatives. These are used to calculate the flow of electricity, model some motion, or set the energy flow of an object in a graphical form.<\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2023-03-18","isAnswered":true,"url":"\/differential-calculus\/103615-how-do-i-find-the-tangent-line","absoluteUrl":"https:\/\/plainmath.org\/differential-calculus\/103615-how-do-i-find-the-tangent-line","user":{"id":98392,"firstName":"Lorelei Johnston","lastName":"","fullName":"Lorelei Johnston ","shortName":"Lorelei Johnston ","email":"lorelei_johnston_@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":4,"title":"Female 3","src":"\/build\/images\/user-picture\/user-4.png","src2x":"\/build\/images\/user-picture\/user-4@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":"","isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":103604,"h1":"How to find vertical asymptotes using limits?","content":"How to find vertical asymptotes using limits?\n","category":{"id":81,"name":"Differential Calculus","title":"Master Differential Calculus Problems with Our Experts","url":"\/post-secondary\/calculus-and-analysis\/differential-calculus","alias":"differential-calculus","seoText":"<div>One of the most famous and challenging aspects of learning calculus and analysis is dealing with differential calculus equations that are not too complex per se if you have an example of differential calculus sample problems that contain both answers and questions. It is recommended to use your logic and see how differential calculus problems are formed with the help of several equations where you must be <a href=\"https:\/\/plainmath.org\/post-secondary\/calculus-and-analysis\/integral-calculus\/decimals\">solving equations with decimals<\/a>, functions and their derivatives. These are used to calculate the flow of electricity, model some motion, or set the energy flow of an object in a graphical form.<\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2023-03-16","isAnswered":true,"url":"\/differential-calculus\/103604-how-to-find-vertical-asymptote","absoluteUrl":"https:\/\/plainmath.org\/differential-calculus\/103604-how-to-find-vertical-asymptote","user":{"id":98372,"firstName":"Ean Chapman","lastName":"","fullName":"Ean Chapman ","shortName":"Ean Chapman ","email":"ean_chapman_@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":5,"title":"Female 4","src":"\/build\/images\/user-picture\/user-5.png","src2x":"\/build\/images\/user-picture\/user-5@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":"","isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":103588,"h1":"How to find the center and radius of the circle x^2-12x+y^2+4y+15=0?","content":"How to find the center and radius of the circle <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"> <mstyle displaystyle=\"true\"> <msup> <mi>x<\/mi> <mn>2<\/mn> <\/msup> <mo>-<\/mo> <mn>12<\/mn> <mi>x<\/mi> <mo>+<\/mo> <msup> <mi>y<\/mi> <mn>2<\/mn> <\/msup> <mo>+<\/mo> <mn>4<\/mn> <mi>y<\/mi> <mo>+<\/mo> <mn>15<\/mn> <mo>=<\/mo> <mn>0<\/mn> <\/mstyle><\/math>?\n","category":{"id":81,"name":"Differential Calculus","title":"Master Differential Calculus Problems with Our Experts","url":"\/post-secondary\/calculus-and-analysis\/differential-calculus","alias":"differential-calculus","seoText":"<div>One of the most famous and challenging aspects of learning calculus and analysis is dealing with differential calculus equations that are not too complex per se if you have an example of differential calculus sample problems that contain both answers and questions. It is recommended to use your logic and see how differential calculus problems are formed with the help of several equations where you must be <a href=\"https:\/\/plainmath.org\/post-secondary\/calculus-and-analysis\/integral-calculus\/decimals\">solving equations with decimals<\/a>, functions and their derivatives. These are used to calculate the flow of electricity, model some motion, or set the energy flow of an object in a graphical form.<\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2023-03-16","isAnswered":true,"url":"\/differential-calculus\/103588-how-to-find-the-center-and-rad","absoluteUrl":"https:\/\/plainmath.org\/differential-calculus\/103588-how-to-find-the-center-and-rad","user":{"id":98342,"firstName":"Jefferson Simmons","lastName":"","fullName":"Jefferson Simmons ","shortName":"Jefferson Simmons ","email":"jefferson_simmons_@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":7,"title":"Male 3","src":"\/build\/images\/user-picture\/user-7.png","src2x":"\/build\/images\/user-picture\/user-7@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":"","isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":103437,"h1":"Let f be a function so that (below). Which must be true? I. f is continuous at x=2 II. f is differentiable at x=2 III. The derivative of f is continuous at x=2 (A) I (B) II (C) I and II (D) I and III (E) II and III","content":"Let f be a function so that (below). Which must be true?<br>I. f is continuous at x=2<br>II. f is differentiable at x=2<br>III. The derivative of f is continuous at x=2<br>(A) I (B) II (C) I and II (D) I and III (E) II and III\n","category":{"id":81,"name":"Differential Calculus","title":"Master Differential Calculus Problems with Our Experts","url":"\/post-secondary\/calculus-and-analysis\/differential-calculus","alias":"differential-calculus","seoText":"<div>One of the most famous and challenging aspects of learning calculus and analysis is dealing with differential calculus equations that are not too complex per se if you have an example of differential calculus sample problems that contain both answers and questions. It is recommended to use your logic and see how differential calculus problems are formed with the help of several equations where you must be <a href=\"https:\/\/plainmath.org\/post-secondary\/calculus-and-analysis\/integral-calculus\/decimals\">solving equations with decimals<\/a>, functions and their derivatives. These are used to calculate the flow of electricity, model some motion, or set the energy flow of an object in a graphical form.<\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2023-03-13","isAnswered":true,"url":"\/differential-calculus\/103437-let-f-be-a-function-so-that-b","absoluteUrl":"https:\/\/plainmath.org\/differential-calculus\/103437-let-f-be-a-function-so-that-b","user":{"id":98020,"firstName":"venomx8e0b","lastName":"","fullName":"venomx8e0b ","shortName":"venomx8e0b ","email":"venomx8e0b_@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":1,"title":"Female 1","src":"\/build\/images\/user-picture\/user-1.png","src2x":"\/build\/images\/user-picture\/user-1@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":"","isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":103461,"h1":"How to find the center and radius of the circle given x^2+y^2+8x-6y=0?","content":"How to find the center and radius of the circle given <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"> <mstyle displaystyle=\"true\"> <msup> <mi>x<\/mi> <mn>2<\/mn> <\/msup> <mo>+<\/mo> <msup> <mi>y<\/mi> <mn>2<\/mn> <\/msup> <mo>+<\/mo> <mn>8<\/mn> <mi>x<\/mi> <mo>-<\/mo> <mn>6<\/mn> <mi>y<\/mi> <mo>=<\/mo> <mn>0<\/mn> <\/mstyle><\/math>?\n","category":{"id":81,"name":"Differential Calculus","title":"Master Differential Calculus Problems with Our Experts","url":"\/post-secondary\/calculus-and-analysis\/differential-calculus","alias":"differential-calculus","seoText":"<div>One of the most famous and challenging aspects of learning calculus and analysis is dealing with differential calculus equations that are not too complex per se if you have an example of differential calculus sample problems that contain both answers and questions. It is recommended to use your logic and see how differential calculus problems are formed with the help of several equations where you must be <a href=\"https:\/\/plainmath.org\/post-secondary\/calculus-and-analysis\/integral-calculus\/decimals\">solving equations with decimals<\/a>, functions and their derivatives. These are used to calculate the flow of electricity, model some motion, or set the energy flow of an object in a graphical form.<\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2023-03-13","isAnswered":true,"url":"\/differential-calculus\/103461-how-to-find-the-center-and-rad","absoluteUrl":"https:\/\/plainmath.org\/differential-calculus\/103461-how-to-find-the-center-and-rad","user":{"id":97978,"firstName":"Shayla Phelps","lastName":"","fullName":"Shayla Phelps ","shortName":"Shayla Phelps ","email":"shayla_phelps_@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":1,"title":"Female 1","src":"\/build\/images\/user-picture\/user-1.png","src2x":"\/build\/images\/user-picture\/user-1@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":"","isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":103455,"h1":"How to find the center and radius of the circle x^2+y^2+4x-8y+4=0?","content":"How to find the center and radius of the circle <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"> <mstyle displaystyle=\"true\"> <msup> <mi>x<\/mi> <mn>2<\/mn> <\/msup> <mo>+<\/mo> <msup> <mi>y<\/mi> <mn>2<\/mn> <\/msup> <mo>+<\/mo> <mn>4<\/mn> <mi>x<\/mi> <mo>-<\/mo> <mn>8<\/mn> <mi>y<\/mi> <mo>+<\/mo> <mn>4<\/mn> <mo>=<\/mo> <mn>0<\/mn> <\/mstyle><\/math>?\n","category":{"id":81,"name":"Differential Calculus","title":"Master Differential Calculus Problems with Our Experts","url":"\/post-secondary\/calculus-and-analysis\/differential-calculus","alias":"differential-calculus","seoText":"<div>One of the most famous and challenging aspects of learning calculus and analysis is dealing with differential calculus equations that are not too complex per se if you have an example of differential calculus sample problems that contain both answers and questions. It is recommended to use your logic and see how differential calculus problems are formed with the help of several equations where you must be <a href=\"https:\/\/plainmath.org\/post-secondary\/calculus-and-analysis\/integral-calculus\/decimals\">solving equations with decimals<\/a>, functions and their derivatives. These are used to calculate the flow of electricity, model some motion, or set the energy flow of an object in a graphical form.<\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2023-03-11","isAnswered":true,"url":"\/differential-calculus\/103455-how-to-find-the-center-and-rad","absoluteUrl":"https:\/\/plainmath.org\/differential-calculus\/103455-how-to-find-the-center-and-rad","user":{"id":98281,"firstName":"Franco Avila","lastName":"","fullName":"Franco Avila ","shortName":"Franco Avila ","email":"franco_avila_@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":5,"title":"Female 4","src":"\/build\/images\/user-picture\/user-5.png","src2x":"\/build\/images\/user-picture\/user-5@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":"","isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":103321,"h1":"How to identify the center and radius of the circle (x+3)^2+(y-8)^2=16?","content":"How to identify the center and radius of the circle <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"> <mstyle displaystyle=\"true\"> <msup> <mrow> <mo>(<\/mo> <mi>x<\/mi> <mo>+<\/mo> <mn>3<\/mn> <mo>)<\/mo> <\/mrow> <mn>2<\/mn> <\/msup> <mo>+<\/mo> <msup> <mrow> <mo>(<\/mo> <mi>y<\/mi> <mo>-<\/mo> <mn>8<\/mn> <mo>)<\/mo> <\/mrow> <mn>2<\/mn> <\/msup> <mo>=<\/mo> <mn>16<\/mn> <\/mstyle><\/math>?\n","category":{"id":81,"name":"Differential Calculus","title":"Master Differential Calculus Problems with Our Experts","url":"\/post-secondary\/calculus-and-analysis\/differential-calculus","alias":"differential-calculus","seoText":"<div>One of the most famous and challenging aspects of learning calculus and analysis is dealing with differential calculus equations that are not too complex per se if you have an example of differential calculus sample problems that contain both answers and questions. It is recommended to use your logic and see how differential calculus problems are formed with the help of several equations where you must be <a href=\"https:\/\/plainmath.org\/post-secondary\/calculus-and-analysis\/integral-calculus\/decimals\">solving equations with decimals<\/a>, functions and their derivatives. These are used to calculate the flow of electricity, model some motion, or set the energy flow of an object in a graphical form.<\/div>","parent":{"id":63,"name":"Calculus and Analysis","title":"Calculus Application Problems: Practice Here","url":"\/post-secondary\/calculus-and-analysis","alias":"calculus-and-analysis","seoText":"College calculus is one of the most frequent subjects that you may encounter. There are certain equations that will sound cryptic as you are looking for answers. The majority of college calculus problems will look odd only because they are not applied to a real-life situation. Take a look at your questions and compare them with the answers that we have presented for you. By doing so, you will approach college-level calculus differently as we have amazing online college calculus samples. If things still seem challenging to you, our college calculus help below will provide you with more understanding.","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. 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