If u is a differentiable function of x, how do you find (ddx)(un) if n is an integer? If n is a real number? Give examples.

Step 1Letx=f(t) and y=f(t) be the parametric equations of a given curve.differentiate the parametric equations with respect to t we havedxdt=f′(t) and dydt=g′(t)Nowdydx=dydt×dtdx=g′(t)f′(t)dydx is the slope of the curve.Now we can expect d2ydx2 if f;(t)≠0That is dxdt≠0Step 2For example,Letx=t−t2and y=t−t3be the parametric equations of the curvedxdt=1−2tand dydt=1−3t2dydx=dydt×dtdx=1−3t21−2tNow,ddx(dydx)=ddx(1−3t21−2t)ddx(dydx)=ddt(1−3t21−2t)×(dtdx)

Step by step solution to "If u is a differentiable function of x,..."

zachutnat4o

Answered question

2021-12-07

If u is a differentiable function of x, how do you find

(\frac{d}{d x}) (u n)

if n is an integer? If n is a real number? Give examples.

Answer & Explanation

Greg Snyder

Beginner2021-12-08Added 11 answers

Step 1
Let
$x = f (t)$ and $y = f (t)$ be the parametric equations of a given curve.
differentiate the parametric equations with respect to t we have
$\frac{d x}{d t} = f^{'} (t)$ and $\frac{d y}{d t} = g^{'} (t)$
Now
$\frac{d y}{d x} = \frac{d y}{d t} \times \frac{d t}{d x}$
$= \frac{g^{'} (t)}{f^{'} (t)}$
$\frac{d y}{d x}$ is the slope of the curve.
Now we can expect $\frac{d^{2} y}{{d x}^{2}}$ if $f; (t) \neq 0$
That is $\frac{d x}{d t} \neq 0$
Step 2
For example,
Let
$x = t - t^{2}$
and $y = t - t^{3}$
be the parametric equations of the curve
$\frac{d x}{d t} = 1 - 2 t$
and $\frac{d y}{d t} = 1 - 3 t^{2}$
$\frac{d y}{d x} = \frac{d y}{d t} \times \frac{d t}{d x}$
$= \frac{1 - 3 t^{2}}{1 - 2 t}$
Now,
$\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{1 - 3 t^{2}}{1 - 2 t})$
$\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d t} (\frac{1 - 3 t^{2}}{1 - 2 t}) \times (\frac{d t}{d x})$

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