The derivative of a composite function $( f\circ g) ( x)$ can be found using the ______ Rule.

True or False If $y=f( u)$ and $u=g( x)$ are differentiable functions, then $y=f( g( x) )$ is differentiable.

True or False If $y=f( g( x) )$ is a differentiable function, then $y^\prime =$ $f^\prime ( g^\prime ( x)) .$

To find the derivative of $y=\tan (1+\cos x)$, using the Chain Rule, begin with $y$ = ______ and $u$ = ______.

If $y=( x^{3}+4x+1) ^{100}$, then $y^\prime =$ ______.

If $f( x) =e^{3x^{2}+5}$, then $f^\prime (x) =$ ______.

True or False The Chain Rule can be applied to multiple composite functions.

$\dfrac{d}{dx}\sin x^{2}=$ ______.

$y=u^{5}$, $u=x^{3}+1$

$y=u^{3}$, $u=2x+5$

${y=\dfrac{u}{u+1}}$, ${u=x^{2}+1}$

${y=\dfrac{u-1}{u}}$, ${u=x^{2}-1}$

${y=(u+1)^{2}}$, ${u=\dfrac{1}{x}}$

${y=(u^{2}-1)^{3}}$, ${u=\dfrac{1}{x+2}}$

$f(x)=(3x+5)^{2}$

$f(x)=(2x-5)^{3}$

$f(x)=(6x-5)^{-3}$

$f(t)=(4t+1)^{-2}$

$g(x)=(x^{2}+5)^{4}$

$F(x)=(x^{3}-2)^{5}$

$f(u)=\left( u-\dfrac{1}{u}\right)^{3}$

${f(x)=\left( {x+\dfrac{1}{x}}\right) ^{3}}$

$g(x) =( 4x+e^{x}) ^{3}$

$F(x) =( e^{x}-x^{2}) ^{2}$

$f(x)=\tan ^{2}x$

$f(x)=\sec ^{3}x$

$f( z) =( \tan z+\cos z) ^{2}$

$f( z) =( e^{z}+2\sin z) ^{3}$

$y=(x^{2}+4)^{2}(2x^{3}-1)^{3}$

$y=(x^{2}-2)^{3}(3x^{4}+1)^{2}$

$y=\left( \dfrac{\sin x}{x}\right) ^{2}$

$y=\left( \dfrac{x+\cos x}{x}\right) ^{5}$

$y=\sin ( 4x)$

$y=\cos (5x)$

$y=2\sin (x^{2}+2x-1)$

$y=\dfrac{1}{2}\cos (x^{3}-2x+5)$

$y=\sin \dfrac{1}{x}$

$y=\sin \dfrac{3}{x}$

$y=\sec ( 4x)$

$y=\cot (5x)$

$y=e^{1/x}$

$y=e^{1/x^{2}}$

$y=\dfrac{1}{x^{4}-2x+1}$

$y=\dfrac{3}{x^{5}+2x^{2}-3}$

$y=\dfrac{100}{1+99e^{-x}}$

$y=\dfrac{1}{1+2e^{-x}}$

$y=2^{\sin x}$

$y=(\sqrt{3}) ^{\cos x}$

$y=6^{\sec x}$

$y=3^{\tan x}$

$y=5xe^{3x}$

$y=x^{3}\,e^{2x}$

$y=x^{2}\sin (4x)$

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

$y=e^{-ax}\sin (bx)$

$y=e^{ax}\cos (-bx)$

$y=\dfrac{e^{ax}-1}{e^{ax}+1}$

$y=\dfrac{e^{-ax}+1}{e^{bx}-1}$

$y=u^{3}$, $u=3v^{2}+1$, $v=\dfrac{4}{x^{2}}$

$y=3u$, $u=3v^{2}-4$, $v=\dfrac{1}{x}$

${y=u^{2}+1}$, ${u=\dfrac{4}{v}},$ ${v=x^{2}}$

${y=u^{3}-1}$, ${u=-\dfrac{2}{v}}$, ${v=x^{3}}$

$y=e^{-2x}\cos (3x)$

$y=e^{\pi x}\tan (\pi x)$

$y=\cos(e^{x^{2}})$

$y=\tan(e^{x^{2}})$

$y=e^{\cos (4x)}$

$y=e^{\csc ^{2}x}$

$y=4\sin ^{2}(3x)$

$y=2\cos ^{2}(x^{2})$

$y=(x^{3}+1)^{2}$

$y=(x^{2}-2)^{3}$

$f(x) =(x^{2}-2x+1)^{5}$ at $(1,0)$

$f(x) =(x^{3}-x^{2}+x-1)^{10}$ at $(0,1)$

$f(x) {=\dfrac{x}{(x^{2}-1)^{3}}}$ at ${\left({2,\dfrac{2}{27}}\right)}$

$f(x) {=\dfrac{x^{2}}{(x^{2}-1)^{2}}}$ at ${\left(2,\dfrac{4}{9}\right) }$

$f( x) =\sin (2x) +\cos \dfrac{x}{2}$ at $(0,1)$

$f(x) =\sin ^{2}x+\cos ^{3}x$ at $\left( \dfrac{\pi }{2},1\right)$

${\dfrac{d\,^{2}}{dx^{2}}\cos (x^{5})}$

${\dfrac{d\,^{3}}{dx^{3}}\sin ^{3}x}$

Suppose $h=f\circ g$. Find $h^\prime (1)$ if $f^\prime (2)=6,$ $f(1)=4,$ $g(1)=2$, and $g^\prime (1)=-2$.

Suppose $h=f\circ g$. Find $h^\prime (1)$ if $f^\prime (3)=4,$ $f(1)=1,$ $g(1)=3$, and $g^\prime (1)=3$.

Suppose $h=g\circ f$. Find $h^\prime (0)$ if $f(0)=3,$ $f^\prime (0)=-1,$ $g(3)=8$, and $g^\prime (3)=0$.

Suppose $h=g\circ f$. Find $h^\prime (2)$ if $f(1)=2,$ $f^\prime (1)=4,$ $f(2)=-3,$ $f^\prime (2)=4,\,\ g(-3)=1$, and $g^\prime (-3)=3$.

If $y=u^{5}+u$ and $u=4x^{3}+x-4$, find $\dfrac{dy}{dx}$ at $x=1$.

If $y=e^{u}+3u$ and $u=\cos x$, find $\dfrac{dy}{dx}$ at $x=0$.

${\dfrac{d}{dx} f(x^{2}\,{+}\,1)}$ (Hint: Let $u\,{=}\,x^{2}\,{+}\,1$.)

${\dfrac{d}{dx} f(1-x^{2})\qquad }$

$\dfrac{d}{dx} f \!\left(\dfrac{x+1}{x-1}\right)$

${\dfrac{d}{dx} f\!\left(\dfrac{1-x}{1+x}\right)}$

${\dfrac{d}{dx} f(\sin x)}$

${\dfrac{d}{dx}f(\tan x)}$

${\dfrac{d^{2}}{dx^{2}} f(\cos x)}$

$\dfrac{d^{2}}{dx^{2}} f(e^{x})$

Rectilinear Motion The distance $s$, in meters, of an object from the origin at time $t ≥ 0$ seconds is given by $s=s( t) = A\cos (\omega t+\phi$), where $A,$ $\omega ,$ and $\phi$ are constant.

Rectilinear Motion A bullet is fired horizontally into a bale of paper. The distance $s$ (in meters) the bullet travels into the bale of paper in $t$ seconds is given by $$s=s( t) =8-(2-t)^{3},\quad 0 ≤ t≤ 2.$$

Rectilinear Motion Find the acceleration $a$ of a car if the distance $s$, in feet, it has traveled along a highway at time $t≥ 0$ seconds is given by $$s( t) =\frac{80}{3}\left[ {{t+\frac{3}{\pi }\sin }}\left( {{\frac{\pi }{6}t}}\right) \right]$$

Rectilinear Motion An object moves in rectilinear motion so that at time $t≥ 0$ seconds, its distance from the origin is $s(t) =\sin e^{t}$, in feet.

Resistance The resistance $R$ (measured in ohms) of an 80-meter-long electric wire of radius $x$ (in centimeters) is given by the formula $R=R(x) =\dfrac{0.0048}{x^{2}}$. The radius $x$ is given by $x=0.1991 + 0.000003T$ where $T$ is the temperature in Kelvin. How fast is $R$ changing with respect to $T$ when $T=320{\,{\rm{K}}}$?

Pendulum Motion in a Car The motion of a pendulum swinging in the direction of motion of a car moving at a low, constant speed, can be modeled by $$s=s(t)=0.05\sin (2t)+3t \ \ \ \ \ \ 0≤ t≤ \pi$$ where $s$ is the distance in meters and $t$ is the time in seconds.

Source: Mathematics students at Trine University.

Economics The function $A(t) =102-90\,e^{-0.21t}$ represents the relationship between $A$, the percentage of the market penetrated by second-generation smart phones, and $t$, the time in years, where $t=0$ corresponds to the year 2010.

Meteorology The atmospheric pressure at a height of $x$ meters above sea level is $P(x)=10^{4}e^{-0.00012x}$ kilograms per square meter. What is the rate of change of the pressure with respect to the height at $x=500$ m? At $x=750$ m?

Hailstones Hailstones originate at an altitude of about $3000{\,{\rm{m}}}$, although this varies. As they fall, air resistance slows down the hailstones considerably. In one model of air resistance, the speed of a hailstone of mass $m$ as a function of time $t$ is given by $v(t)=\dfrac{mg}{k}( 1-e^{-kt/m}) {\,{\rm{m}}}/\!{\,{\rm{s}}}$, where $g=9.8{\,{\rm{m}}}/\!{\,{\rm{s}}}^{2}$ is the acceleration due to gravity and $k$ is a constant that depends on the size of the hailstone and the conditions of the air.

Mean Earnings The mean earnings $E$, in dollars, of workers 18 years and over are given in the table below:

Rectilinear Motion An object moves in rectilinear motion so that at time $t>0$ its distance $s$ from the origin is $s=s( t) .$ The velocity $v$ of the object is $v=\dfrac{ds}{dt},$ and its acceleration is $a=\dfrac{dv}{dt}=\dfrac{d^{2}s}{dt^{2}}.$ If the velocity $v=v ( s)$ is expressed as a function of $s$, show that the acceleration $a$ can be expressed as $a=v\dfrac{dv}{ds}.$

Student Approval Professor Miller’s student approval rating is modeled by the function $Q( t) =21+\dfrac{10\sin \left(\dfrac{2\pi t}{7}\right) }{\sqrt{t}-\sqrt{20}},$ where $0≤ t≤ 16$ is the number of weeks since the semester began.

Source: Mathematics students at Millikin University, Decatur, Illinois.

Angular Velocity If the disk in the figure is rotated about the vertical through an angle $\theta$, torsion in the wire attempts to turn the disk in the opposite direction. The motion $\theta$ at time $t$ (assuming no friction or air resistance) obeys the equation $$\theta ( t) =\dfrac{\pi }{3}\cos \left( \dfrac{1}{2}\sqrt{\dfrac{2k}{5}}t\right)$$ where $k$ is the coefficient of torsion of the wire.

Harmonic Motion A weight hangs on a spring making it $2{\,{\rm{m}}}$ long when it is stretched out (see the figure). If the weight is pulled down and then released, the weight oscillates up and down, and the length $l$ of the spring after $t$ seconds is given by $l( t) =2+\cos \left(2\pi t\right)$.

Find $F^\prime (1)$ if $f(x)=\sin x$ and $F(t)=f(t^{2}-1)$.

Normal Line Find the point on the graph of $y=e^{-x}$ where the normal line to the graph passes through the origin.

Use the Chain Rule and the fact that ${\cos x=\sin \left(\dfrac{\pi }{2}-x\right) }$ to show that $\dfrac{d}{dx}\cos x=-\sin x.$

If $y=e^{2x}$, show that $y^{\prime\prime} -4y=0$.

If $y=e^{-2x}$, show that $y^{\prime\prime} -4y=0$.

If $y=Ae^{2x}+Be^{-2x}$, where $A$ and $B$ are constants, show that $y^{\prime\prime} -4y=0$.

If $y=Ae^{ax}+Be^{-ax}$, where $A$, $B$, and $a$ are constants, show that $y^{\prime\prime} -a^{2}y=0$.

If $y=Ae^{2x}+Be^{3x}$, where $A$ and $B$ are constants, show that $y^{\prime\prime} -5y^\prime +6y=0$.

If $y=Ae^{-2x}+Be^{-x}$, where $A$ and $B$ are constants, show that $y^{\prime\prime} +3y^\prime +2y=0$.

If $y=A~\sin ( \omega t) +B~\cos (\omega t)$, where $A, B$, and $\omega$ are constants, show that $y^{\prime \prime }+\omega ^{2}y=0$.

Show that $\dfrac{d}{dx}f(h(x))=2xg(x^{2})$, if $\dfrac{d}{dx}f(x)=g(x)$ and $h(x)=x^{2}$.

Find the $n$th derivative of $f(x)=(2x+3)^{n}$.

Find a general formula for the $n$th derivative of $y$.

If $y=e^{-at}~[ A~\sin ( \omega t) +B~\cos ( \omega t) ]$, where $A$, $B$, $a$, and $\omega$ are constants, find $y^\prime$ and $y^{\prime\prime}$.

Show that if a function $f$ has the properties:

• $f(u+v)=f(u)f(v)$ for all choices of $u$ and $v$
• $f(x)=1+xg(x)$, where $\lim\limits_{x\rightarrow 0}g(x)=1,$

then $f^\prime =f$.

Find the $n$th derivative of $f(x)=\dfrac{1}{3x-4}$.

Let $f_{1}(x),\ldots ,f_{n}(x)$ be $n$ differentiable functions. Find the derivative of $y=f_{1}(f_{2}(f_{3}$ $(\ldots (f_{n}(x)\ldots ))))$.

Let $$f(x)=\left\{ \begin{array}{c@{\qquad}ll} x^{2}\sin {\dfrac{1}{x}} & \hbox{if} & x\;≠ 0 \\ 0 & \hbox{if} & x\;=0 \end{array} \right.$$

Show that $f^\prime (0)$ exists, but that $f^\prime (x)$ is not continuous at $0$.

Define $f$ by $$f(x)=\left\{ \begin{array}{l@{\quad}ll} e^{-1/x^{2}} & \hbox{if}& x\;≠ 0 \\ 0 & \hbox{if}& x\;=0 \end{array} \right.$$

Show that $f$ is differentiable on $(-\infty ,\infty )$ and find $f^\prime (x)$ for each value of $x$. [Hint: To find $f^\prime (0)$, use the definition of the derivative. Then show that $1< x^{2}e^{1/x^{2}}$ for $x≠ 0$.]

Suppose $f( x) =x^{2}$ and $g( x) =\left\vert x-1\right\vert .$ The functions $f$ and $g$ are continuous on their domains, the set of all real numbers.

Suppose $f( x) =x^{4}$ and $g( x) =x^{1/3}.$ The functions $f$ and $g$ are continuous on their domains, the set of all real numbers.

The function $f(x)=e^{x}$ has the property $f^\prime (x)=f(x)$. Give an example of another function $g(x)$ such that $g(x)$ is defined for all real $x$, $g^{\prime }(x)=g(x)$, and $g(x)≠ f(x)$.

Harmonic Motion The motion of the piston of an automobile engine is approximately simple harmonic. If the stroke of a piston (twice the amplitude) is $10{\,{\rm{cm}}}$ and the angular velocity $\omega$ is $60$ revolutions per second, then the motion of the piston is given by $s( t) =5\sin ( 120\pi t ) {\,{\rm{cm}}}.$