미분법 전체 문제

Find an equation of the tangent line to the graph of the equation ${\tan }^{-1}(x+y)={x}^{2}+\frac{\pi }{4}$ at the point $(0,1).$

For the following exercises, find an equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.

Find ${y}^{'}$ and ${y}^{''}$ for ${x}^{2}+6xy-2{y}^{2}=3.$

[T] The number of cell phones produced when $x$ dollars is spent on labor and $y$ dollars is spent on capital invested by a manufacturer can be modeled by the equation $60{x}^{3/4}{y}^{1/4}=3240.$

a. Find $\frac{dy}{dx}$ and evaluate at the point $(81,16).$ b. Interpret the result of a.

[T] The number of cars produced when $x$ dollars is spent on labor and $y$ dollars is spent on capital invested by a manufacturer can be modeled by the equation $30{x}^{1/3}{y}^{2/3}=360.$

(Both $x$ and $y$ are measured in thousands of dollars.)

a. Find $\frac{dy}{dx}$ and evaluate at the point $(27,8).$ b. Interpret the result of a.

The volume of a right circular cone of radius $x$ and height $y$ is given by $V=\frac{1}{3}\pi {x}^{2}y.$ Suppose that the volume of the cone is a constant. Find $\frac{dy}{dx}$ when $x=4$ and $y=16.$

Find an equation for the surface area of the rectangular box, $S(x,y).$

If the surface area of the rectangular box is 78 square feet, find $\frac{dy}{dx}$ when $x=3$ feet and $y=5$ feet.

For the following exercises, use implicit differentiation to determine ${y}^{'}.$ Does the answer agree with the formulas we have previously determined?

$x=\sin y$

For the following exercises, use implicit differentiation to determine ${y}^{'}.$ Does the answer agree with the formulas we have previously determined?

$x=\cos y$

For the following exercises, use implicit differentiation to determine ${y}^{'}.$ Does the answer agree with the formulas we have previously determined?

$x=\tan y$

For the following exercises, find ${f}^{'}(x)$ for each function.

$f(x)=\frac{{e}^{-x}}{x}$

For the following exercises, find ${f}^{'}(x)$ for each function.

$f(x)={e}^{{x}^{3}\ln x}$

For the following exercises, find ${f}^{'}(x)$ for each function.

$f(x)=\sqrt{{e}^{2x}+2x}$

For the following exercises, find ${f}^{'}(x)$ for each function.

$f(x)=\frac{{e}^{x}-{e}^{-x}}{{e}^{x}+{e}^{-x}}$

For the following exercises, find ${f}^{'}(x)$ for each function.

$f(x)=\frac{{10}^{x}}{\ln 10}$

For the following exercises, find ${f}^{'}(x)$ for each function.

$f(x)={2}^{4x}+4{x}^{2}$

For the following exercises, find ${f}^{'}(x)$ for each function.

$f(x)={3}^{\sin 3\text{x}}$

For the following exercises, find ${f}^{'}(x)$ for each function.

$f(x)=\ln (4{x}^{3}+x)$

For the following exercises, find ${f}^{'}(x)$ for each function.

$f(x)=\ln \sqrt{5x-7}$

For the following exercises, find ${f}^{'}(x)$ for each function.

$f(x)={x}^{2}\ln 9x$