# Calculus 1: Derivatives questions and answers Recent questions in Derivatives
Derivatives
ANSWERED ### Determine the order of the given partial differential equation; also state whether the equation is linear or nonlinear. $$\displaystyle{5}^{{{2}}}{u}_{{{x}}}+{u}_{{{t}}}={1}+{u}_{{\times}}$$

Derivatives
ANSWERED ### Compute the gradient of the following functions and evaluate it at the given point P. $$\displaystyle{g{{\left({x},{y}\right)}}}={x}^{{{2}}}-{4}{x}^{{{2}}}{y}-{8}{x}{y}^{{{2}}};{P}{\left(-{1},{2}\right)}$$

Derivatives
ANSWERED ### The radius of a sphere increases at a rate of $$\displaystyle{4}\frac{{m}}{{\sec{}}}$$. Find the rate at which the surface area increases when the radius is 9m. $$\displaystyle{288}\pi\frac{{m}^{{{2}}}}{{\sec{}}}$$ $$\displaystyle{36}\pi\frac{{m}^{{{2}}}}{{\sec{}}}$$ $$\displaystyle{18}\pi\frac{{m}^{{{2}}}}{{\sec{}}}$$ $$\displaystyle{72}\pi\frac{{m}^{{{2}}}}{{\sec{}}}$$ $$\displaystyle{144}\pi\frac{{m}^{{{2}}}}{{\sec{}}}$$

Derivatives
ANSWERED ### Using the definition, calculate the derivatives of the function. Then find the values of the derivatives as specified. $$\displaystyle{g{{\left({t}\right)}}}={\frac{{{1}}}{{{t}^{{{2}}}}}};{g}'{\left(-{1}\right)},{g}'{\left({2}\right)},{g}'{\left(\sqrt{{{3}}}\right)}$$

Derivatives
ANSWERED ### Using the definition, calculate the derivatives of the functions. Then find the values of the derivatives as specified. $$\displaystyle{k}{\left({z}\right)}={\frac{{{1}-{z}}}{{{2}{z}}}};{k}'{\left(-{1}\right)},{k}'{\left({1}\right)},{k}'{\left(\sqrt{{{2}}}\right)}$$

Derivatives
ANSWERED ### Find the derivatives of $$\displaystyle f{{\left({x}\right)}}=\frac{3}{{2}}{x}^{4}+{2}{x}^{3}-{5}{x}^{2}+{9}$$

Derivatives
ANSWERED ### Find the derivatives of $$\displaystyle{h}{\left({x}\right)}={\left(-{7}{\left({x}^{2}-{8}\right)}{2}\right)}{2}{x}$$

Derivatives
ANSWERED ### Two balls A and B were Initially 9 m apart. Ball A moved at a speed of vA=4 m/s and ball B moved at a speed of vb=8 m/s in the directions as shown in the figure below. Derivatives
ANSWERED ### Need help with the following question Compute the directional derivative of $$\displaystyle f{{\left({x},{y}\right)}}={e}^{2}{x}-{5}{y}$$

Derivatives
ANSWERED ### Define $$\displaystyle g{{\left({x},{y}\right)}}={x}^{2}+{y}^{2}-{4}{x}{y}+{3}{y}+{2}$$

Derivatives
ANSWERED ### Find the derivative $$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{\arctan{{\left({x}^{{2}}+{1}\right)}}}\right]}$$ a) $$\displaystyle{\left({a}{r}{c}{\sec{{\left({x}^{{2}}+{1}\right)}}}\right)}^{{2}}$$ b) $$\displaystyle{\frac{{{1}}}{{{2}+{x}^{{2}}}}}$$ c) $$\displaystyle{\frac{{{1}}}{{{1}+{\left({x}^{{2}}+{1}\right)}^{{2}}}}}$$ d) $$\displaystyle{\frac{{{2}{x}}}{{{1}+{\left({x}^{{2}}+{1}\right)}^{{2}}}}}$$

Derivatives
ANSWERED ### Find the point on the line $$y=4x+3$$ that is closest to the origin. (x,y)=( )

Derivatives
ANSWERED ### $${y}{\left({x}^{2}-{1}\right)}{\left.{d}{y}\right.}+{x}{\left({y}^{2}+{1}\right)}{\left.{d}{x}\right.}={0}$$

Derivatives
ANSWERED ### Find $$\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}$$ and $$\displaystyle\frac{{{d}^{{{2}}}y}}{{\left.{d}{x}\right.}^{{{2}}}}$$. $$\displaystyle{x}={e}^{{{t}}},{y}={t}{e}^{{−{t}}}$$ For which values of t is the curve concave upward?

Derivatives
ANSWERED ### Check whether the given function is satisfying the Fubini’s theorem about ‎mixed partial derivative ‎ $$\displaystyle f{{\left({x},{y}\right)}}={3}{x}^{2} \sin{{\left({4}{x}{y}\right)}}$$

Derivatives
ANSWERED ### If $$\displaystyle{{f}^{{{\left({n}\right)}}}{\left({0}\right)}}={\left({n}+{1}\right)}!\ \text{ for }\ {n}={0},{1},{2},\dot{{s}}$$, find the Maclaurin series forf and its radius of convergence.

Derivatives
ANSWERED ### At what point does the curve have maximum curvature? $$\displaystyle{y}={9}{\ln{{\left({x}\right)}}}$$ What happens to the curvature as $$\displaystyle{x}\rightarrow\infty$$ ?

Derivatives
ANSWERED ### $$\displaystyle{v}={\frac{{{x}\cdot{y}}}{{{x}-{y}}}}$$ Find all of the second partial derivatives

Derivatives
ANSWERED ### how to find the derivative of $$\displaystyle f{{\left({u}\right)}}={9}{e}^{u}+{20}$$
ANSWERED 