합성함수의 미분법

For the following exercises, find $\frac{dy}{dx}$ for each function.

$y={(\tan x+\sin x)}^{-3}$

For the following exercises, find $\frac{dy}{dx}$ for each function.

$y={x}^{2}{\cos }^{4}x$

For the following exercises, find $\frac{dy}{dx}$ for each function.

$y=\sin (\cos 7x)$

For the following exercises, find $\frac{dy}{dx}$ for each function.

$y=\sqrt{6+\sec \pi {x}^{2}}$

For the following exercises, find $\frac{dy}{dx}$ for each function.

$y={\cot }^{3}(4x+1)$

For the following exercises, find $\frac{dy}{dx}$ for each function.

Let $y={[f(x)]}^{2}$ and suppose that ${f}^{'}(1)=4$ and $\frac{dy}{dx}=10$ for $x=1.$ Find $f(1).$

For the following exercises, find $\frac{dy}{dx}$ for each function.

Let $y={(f(x)+5{x}^{2})}^{4}$ and suppose that $f(-1)=-4$ and $\frac{dy}{dx}=3$ when $x=-1.$ Find ${f}^{'}(-1)$

For the following exercises, find $\frac{dy}{dx}$ for each function.

Let $y={(f(u)+3x)}^{2}$ and $u={x}^{3}-2x.$ If $f(4)=6$ and $\frac{dy}{dx}=18$ when $x=2,$ find ${f}^{'}(4).$

[T] Find an equation of the tangent line to $y=-\sin \left(\frac{x}{2}\right)$ at the origin. Use a calculator to graph the function and the tangent line together.

[T] Find an equation of the tangent line to $y={\left(3x+\frac{1}{x}\right)}^{2}$ at the point $(1,16).$ Use a calculator to graph the function and the tangent line together.

Find the $x$-coordinates at which the tangent line to $y={\left(x-\frac{6}{x}\right)}^{8}$ is horizontal.

[T] Find an equation of the line that is normal to $g(\theta )={\sin }^{2}(\pi \theta )$ at the point $\left(\frac{1}{4},\frac{1}{2}\right).$ Use a calculator to graph the function and the normal line together.

For the following exercises, use the information in the following table to find ${h}^{'}(a)$ at the given value for $a.$

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
0	2	5	0	2
1	1	−2	3	0
2	4	4	1	−1
3	3	−3	2	3

$h(x)=f(g(x));a=0$

For the following exercises, use the information in the following table to find ${h}^{'}(a)$ at the given value for $a.$

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
0	2	5	0	2
1	1	−2	3	0
2	4	4	1	−1
3	3	−3	2	3

$h(x)=g(f(x));a=0$

For the following exercises, use the information in the following table to find ${h}^{'}(a)$ at the given value for $a.$

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
0	2	5	0	2
1	1	−2	3	0
2	4	4	1	−1
3	3	−3	2	3

$h(x)={({x}^{4}+g(x))}^{-2};a=1$

For the following exercises, use the information in the following table to find ${h}^{'}(a)$ at the given value for $a.$

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
0	2	5	0	2
1	1	−2	3	0
2	4	4	1	−1
3	3	−3	2	3

$h(x)={\left(\frac{f(x)}{g(x)}\right)}^{2};a=3$

For the following exercises, use the information in the following table to find ${h}^{'}(a)$ at the given value for $a.$

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
0	2	5	0	2
1	1	−2	3	0
2	4	4	1	−1
3	3	−3	2	3

$h(x)=f(x+f(x));a=1$

For the following exercises, use the information in the following table to find ${h}^{'}(a)$ at the given value for $a.$

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
0	2	5	0	2
1	1	−2	3	0
2	4	4	1	−1
3	3	−3	2	3

$h(x)={(1+g(x))}^{3};a=2$

For the following exercises, use the information in the following table to find ${h}^{'}(a)$ at the given value for $a.$

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
0	2	5	0	2
1	1	−2	3	0
2	4	4	1	−1
3	3	−3	2	3

$h(x)=f(g(\sin x));a=0$

[T] The position function of a freight train is given by $s(t)=100{(t+1)}^{-2},$ with $s$ in meters and $t$ in seconds. At time $t=6$ s, find the train’s

a. velocity and b. acceleration. c. Using a. and b. is the train speeding up or slowing down?