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Calculus: Graphical, Numerical, Algebraic

Ross L. Finney, Franklin D. Demana, Bet K. Waits,

Chapter 4

More Derivatives - all with Video Answers

Educators

Section 1

Chain Rule

01:39

Problem 1

Use the given substitution and the Chain Rule to find $d y / d x$.

$$y=\sin (3 x+1), u=3 x+1$$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:12

Problem 2

Use the given substitution and the Chain Rule to find $d y / d x$.

$$y=\sin (7-5 x), u=7-5 x$$

Adrian Co
Adrian Co
Numerade Educator
01:08

Problem 3

Use the given substitution and the Chain Rule to find $d y / d x$.

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

Adrian Co
Adrian Co
Numerade Educator
01:20

Problem 4

Use the given substitution and the Chain Rule to find $d y / d x$.

$$y=\tan \left(2 x-x^{3}\right), u=2 x-x^{3}$$

Adrian Co
Adrian Co
Numerade Educator
02:13

Problem 5

Use the given substitution and the Chain Rule to find $d y / d x$.

$$y=\left(\frac{\sin x}{1+\cos x}\right)^{2}, \quad u=\frac{\sin x}{1+\cos x}$$

Adrian Co
Adrian Co
Numerade Educator
01:19

Problem 6

Use the given substitution and the Chain Rule to find $d y / d x$.

$$y=5 \cot \left(\frac{2}{x}\right), u=\frac{2}{x}$$

Adrian Co
Adrian Co
Numerade Educator
01:12

Problem 7

Use the given substitution and the Chain Rule to find $d y / d x$.

$$y=\cos (\sin x), u=\sin x$$

Adrian Co
Adrian Co
Numerade Educator
01:11

Problem 8

Use the given substitution and the Chain Rule to find $d y / d x$.

$$y=\sec (\tan x), u=\tan x$$

Adrian Co
Adrian Co
Numerade Educator
01:02

Problem 9

An object moves along the $x$ -axis so that its position at any time $t \geq 0$ is given by $x(t)=s(t) .$ Find the velocity of the object as a function of $t$.

$$s=\cos \left(\frac{\pi}{2}-3 t\right)$$

Adrian Co
Adrian Co
Numerade Educator
01:13

Problem 10

An object moves along the $x$ -axis so that its position at any time $t \geq 0$ is given by $x(t)=s(t) .$ Find the velocity of the object as a function of $t$.

$$s=t \cos (\pi-4 t)$$

Adrian Co
Adrian Co
Numerade Educator
01:29

Problem 11

An object moves along the $x$ -axis so that its position at any time $t \geq 0$ is given by $x(t)=s(t) .$ Find the velocity of the object as a function of $t$.

$$s=\frac{4}{3 \pi} \sin 3 t+\frac{4}{5 \pi} \cos 5 t$$

Adrian Co
Adrian Co
Numerade Educator
01:07

Problem 12

An object moves along the $x$ -axis so that its position at any time $t \geq 0$ is given by $x(t)=s(t) .$ Find the velocity of the object as a function of $t$.

$$s=\sin \left(\frac{3 \pi}{2} t\right)+\cos \left(\frac{7 \pi}{4} t\right)$$

Adrian Co
Adrian Co
Numerade Educator
01:08

Problem 13

Find $d y / d x$.

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

Adrian Co
Adrian Co
Numerade Educator
01:48

Problem 14

Find $d y / d x$.

$$y=(\csc x+\cot x)^{-1}$$

Adrian Co
Adrian Co
Numerade Educator
01:15

Problem 15

Find $d y / d x$.

$$y=\sin ^{-5} x-\cos ^{3} x$$

Adrian Co
Adrian Co
Numerade Educator
01:39

Problem 16

Find $d y / d x$.

$$y=x^{3}(2 x-5)^{4}$$

Adrian Co
Adrian Co
Numerade Educator
01:07

Problem 17

Find $d y / d x$.

$$y=\sin ^{3} x \tan 4 x$$

Adrian Co
Adrian Co
Numerade Educator
01:26

Problem 18

Find $d y / d x$.

$$y=4 \sqrt{\sec x+\tan x}$$

Adrian Co
Adrian Co
Numerade Educator
01:40

Problem 19

Find $d y / d x$.

$$y=\frac{3}{\sqrt{2 x+1}}$$

Adrian Co
Adrian Co
Numerade Educator
04:14

Problem 20

Find $d y / d x$.

$$y=\frac{x}{\sqrt{1+x^{2}}}$$

Andrija Isakov
Andrija Isakov
Numerade Educator
01:05

Problem 21

Find $d y / d x$.

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

Adrian Co
Adrian Co
Numerade Educator
01:20

Problem 22

Find $d y / d x$.

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

Adrian Co
Adrian Co
Numerade Educator
01:39

Problem 23

Find $d y / d x$.

$$y=\left(1+\cos ^{2} 7 x\right)^{3}$$

Adrian Co
Adrian Co
Numerade Educator
01:05

Problem 24

Find $d y / d x$.

$$y=\sqrt{\tan 5 x}$$

Adrian Co
Adrian Co
Numerade Educator
01:03

Problem 25

Find $d r / d \theta$.

$$r=\tan (2-\theta)$$

Adrian Co
Adrian Co
Numerade Educator
01:12

Problem 26

Find $d r / d \theta$.

$$r=\sec 2 \theta \tan 2 \theta$$

Adrian Co
Adrian Co
Numerade Educator
03:53

Problem 27

Find $d r / d \theta$.

$$r=\sqrt{\theta \sin \theta}$$

Andrija Isakov
Andrija Isakov
Numerade Educator
01:52

Problem 28

Find $d r / d \theta$.

$$r=2 \theta \sqrt{\sec \theta}$$

Adrian Co
Adrian Co
Numerade Educator
01:15

Problem 29

Find $y^{\prime \prime}$.

$$y=\tan x$$

Adrian Co
Adrian Co
Numerade Educator
01:13

Problem 30

Find $y^{\prime \prime}$.

$$y=\cot x$$

Adrian Co
Adrian Co
Numerade Educator
01:39

Problem 31

Find $y^{\prime \prime}$.

$$y=\cot (3 x-1)$$

Adrian Co
Adrian Co
Numerade Educator
01:33

Problem 32

Find $y^{\prime \prime}$.

$$y=9 \tan (x / 3)$$

Robert Daugherty
Robert Daugherty
Numerade Educator
02:01

Problem 33

Find the value of $(f \circ g)^{\prime}$ at the given value of $x$.

$$f(u)=u^{5}+1, \quad u=g(x)=\sqrt{x}, \quad x=1$$

Robert Daugherty
Robert Daugherty
Numerade Educator
02:08

Problem 34

Find the value of $(f \circ g)^{\prime}$ at the given value of $x$.

$$f(u)=1-\frac{1}{u}, \quad u=g(x)=\frac{1}{1-x}, \quad x=-1$$

Robert Daugherty
Robert Daugherty
Numerade Educator
03:51

Problem 35

Find the value of $(f \circ g)^{\prime}$ at the given value of $x$.

$$f(u)=\cot \frac{\pi u}{10}, \quad u=g(x)=5 \sqrt{x}, \quad x=1$$

Robert Daugherty
Robert Daugherty
Numerade Educator
04:56

Problem 36

Find the value of $(f \circ g)^{\prime}$ at the given value of $x$.

$$f(u)=u+\frac{1}{\cos ^{2} u}, \quad u=g(x)=\pi x, \quad x=\frac{1}{4}$$

Robert Daugherty
Robert Daugherty
Numerade Educator
05:00

Problem 37

Find the value of $(f \circ g)^{\prime}$ at the given value of $x$.

$$f(u)=\frac{2 u}{u^{2}+1}, \quad u=g(x)=10 x^{2}+x+1, \quad x=0$$

Robert Daugherty
Robert Daugherty
Numerade Educator
07:40

Problem 38

Find the value of $(f \circ g)^{\prime}$ at the given value of $x$.

$$f(u)=\left(\frac{u-1}{u+1}\right)^{2}, \quad u=g(x)=\frac{1}{x^{2}}-1, \quad x=-1$$

Robert Daugherty
Robert Daugherty
Numerade Educator
03:42

Problem 39

What happens if you can write a function as a composite in different ways? Do you get the same derivative each time? The Chain Rule says you should. Try it with the functions.

Find $d y / d x$ if $y=\cos (6 x+2)$ by writing $y$ as a composite with
(a) $y=\cos u$ and $u=6 x+2$.
(b) $y=\cos 2 u$ and $u=3 x+1$

Andrija Isakov
Andrija Isakov
Numerade Educator
03:34

Problem 40

What happens if you can write a function as a composite in different ways? Do you get the same derivative each time? The Chain Rule says you should. Try it with the functions.

Find $d y / d x$ if $y=\sin \left(x^{2}+1\right)$ by writing $y$ as a composite with
(a) $y=\sin (u+1)$ and $u=x^{2}$.
(b) $y=\sin u$ and $u=x^{2}+1$

Andrija Isakov
Andrija Isakov
Numerade Educator
05:17

Problem 41

Find the equation of the line tangent to the curve at the point defined by the given value of $t$.

$$x=2 \cos t, \quad y=2 \sin t, \quad t=\pi / 4$$

Andrija Isakov
Andrija Isakov
Numerade Educator
07:05

Problem 42

Find the equation of the line tangent to the curve at the point defined by the given value of $t$.

$$x=\sin 2 \pi t, \quad y=\cos 2 \pi t, \quad t=-1 / 6$$

Andrija Isakov
Andrija Isakov
Numerade Educator
10:25

Problem 43

Find the equation of the line tangent to the curve at the point defined by the given value of $t$.

$$x=\sec ^{2} t-1, \quad y=\tan t, \quad t=-\pi / 4$$

Andrija Isakov
Andrija Isakov
Numerade Educator
05:10

Problem 44

Find the equation of the line tangent to the curve at the point defined by the given value of $t$.

$$x=\sec t, \quad y=\tan t, \quad t=\pi / 6$$

Andrija Isakov
Andrija Isakov
Numerade Educator
05:07

Problem 45

Find the equation of the line tangent to the curve at the point defined by the given value of $t$.

$$x=t, \quad y=\sqrt{t,} \quad t=1 / 4$$

Andrija Isakov
Andrija Isakov
Numerade Educator
04:32

Problem 46

Find the equation of the line tangent to the curve at the point defined by the given value of $t$.

$$x=2 t^{2}+3, \quad y=t^{4}, \quad t=-1$$

Andrija Isakov
Andrija Isakov
Numerade Educator
06:18

Problem 47

Find the equation of the line tangent to the curve at the point defined by the given value of $t$.

$$x=t-\sin t, \quad y=1-\cos t, \quad t=\pi / 3$$

Andrija Isakov
Andrija Isakov
Numerade Educator
04:48

Problem 48

Find the equation of the line tangent to the curve at the point defined by the given value of $t$.

$$x=\cos t, \quad y=1+\sin t, \quad t=\pi / 2$$

Andrija Isakov
Andrija Isakov
Numerade Educator
06:32

Problem 49

Let $x=t^{2}+t,$ and let $y=\sin t$
(a) Find $d y / d x$ as a function of $t$.
(b) Find $\frac{d}{d t}\left(\frac{d y}{d x}\right)$ as a function of $t$
(c) Find $\frac{d}{d x}\left(\frac{d y}{d x}\right)$ as a function of $t$.
Use the Chain Rule and your answer from part (b).
(d) Which of the expressions in parts (b) and (c) is $d^{2} y / d x^{2} ?$

Andrija Isakov
Andrija Isakov
Numerade Educator
02:29

Problem 50

A circle of radius 2 and center (0,0) can be parametrized by the equations $x=2 \cos t$ and $y=2 \sin t .$ Show that for any value of $t,$ the line tangent to the circle at $(2 \cos t, 2 \sin t)$ is perpendicular to the radius.

Bahar Tehranipoor
Bahar Tehranipoor
Numerade Educator
01:08

Problem 51

Let $s=\cos \theta .$ Evaluate $d s / d t$ when $\theta=3 \pi / 2$ and $d \theta / d t=5$.

Linh Vu
Linh Vu
Numerade Educator
01:15

Problem 52

Let $y=x^{2}+7 x-5 .$ Evaluate $d y / d t$ when $x=1$ and $d x / d t=1 / 3$.

Bahar Tehranipoor
Bahar Tehranipoor
Numerade Educator
00:56

Problem 53

What is the largest value possible for the slope of the curve $y=\sin (x / 2) ?$

Linh Vu
Linh Vu
Numerade Educator
01:22

Problem 54

Write an equation for the tangent to the curve $y=\sin m x$ at the origin.

Bahar Tehranipoor
Bahar Tehranipoor
Numerade Educator
04:06

Problem 55

Find the lines that are tangent and normal to the curve $y=2 \tan (\pi x / 4)$ at $x=1 .$ Support your answer graphically.

Linh Vu
Linh Vu
Numerade Educator
15:20

Problem 56

Suppose that functions $f$ and $g$ and their derivatives have the following values at $x=2$ and $x=3$.
$$\begin{array}{c|cccc}
x & f(x) & g(x) & f^{\prime}(x) & g^{\prime}(x) \\
\hline 2 & 8 & 2 & 1 / 3 & -3 \\
3 & 3 & -4 & 2 \pi & 5
\end{array}$$
Evaluate the derivatives with respect to $x$ of the following combinations at the given value of $x$.
(a) $2 f(x)$ at $x=2$
(b) $f(x)+g(x)$ at $x=3$
(c) $f(x) \cdot g(x)$ at $x=3$
(d) $f(x) / g(x)$ at $x=2$
(e) $f(g(x))$ at $x=2$
(f) $\sqrt{f(x)}$ at $x=2$
(g) $1 / g^{2}(x)$ at $x=3$
(h) $\sqrt{f^{2}(x)+g^{2}(x)}$ at $x=2$

Robert Daugherty
Robert Daugherty
Numerade Educator
04:28

Problem 57

Show that $\frac{d}{d x} \cos \left(x^{\circ}\right)$ is $-\frac{\pi}{180} \sin \left(x^{\circ}\right)$.

Andrija Isakov
Andrija Isakov
Numerade Educator
16:09

Problem 58

(a) $5 f(x)-g(x), \quad x=1$
(b) $f(x) g^{3}(x), \quad x=0$
(c) $\frac{f(x)}{g(x)+1}, \quad x=1$
(d) $f(g(x)), \quad x=0$
(e) $g(f(x)), \quad x=0$
(f) $(g(x)+f(x))^{-2}, \quad x=1$
(g) $f(x+g(x)), \quad x=0$

Andrija Isakov
Andrija Isakov
Numerade Educator
01:06

Problem 59

Two curves are said to cross at right angles if their tangents are perpendicular at the crossing point. The technical word for "crossing at right angles" is orthogonal. Show that the curves $y=\sin 2 x$ and $y=-\sin (x / 2)$ are orthogonal at the origin. Draw both graphs and both tangents in a square viewing window.

Linh Vu
Linh Vu
Numerade Educator
01:02

Problem 60

Explain why the Chain Rule formula
$$\frac{d y}{d x}=\frac{d y}{d u} \cdot \frac{d u}{d x}$$
is not simply the well-known rule for multiplying fractions.

Carson Merrill
Carson Merrill
Numerade Educator
05:45

Problem 61

Running Machinery Too Fast Suppose that a piston is moving straight up and down and that its position at time $t$ seconds is
$$s=A \cos (2 \pi b t)$$
with $A$ and $b$ positive. The value of $A$ is the amplitude of the motion, and $b$ is the frequency (number of times the piston moves up and down each second). What effect does doubling the frequency have on the piston's velocity, acceleration, and jerk? (Once you find out, you will know why machinery breaks when you run it too fast.)

Andrija Isakov
Andrija Isakov
Numerade Educator
01:18

Problem 62

The graph in Figure 4.6 shows the variation in average daily temperature in Tempe, Arizona, during a typical 365 -day year. The equation that approximates the Fahrenheit temperature on day $x$ is
$$y=19.3 \sin \left[\frac{2 \pi}{365}(x-101)\right]+70$$
(a) On approximately what day of the year does the daily temperature show the greatest increase from the previous day?
(b) About how many degrees per day is the temperature increasing at that time of the year?

Carson Merrill
Carson Merrill
Numerade Educator
07:05

Problem 63

The position of a particle moving along a coordinate line is $s=\sqrt{1}+4 t,$ with $s$ in meters and $t$ in seconds. Find the particle's velocity and acceleration at $t=6 \mathrm{sec}$.

Ian Grigsby
Ian Grigsby
Numerade Educator
04:04

Problem 64

Suppose the velocity of a falling body is $v=k \vee s \mathrm{~m} / \mathrm{sec}(k$ a constant $)$ at the instant the body has fallen $s$ meters from its starting point. Show that the body's acceleration is constant.

Andrija Isakov
Andrija Isakov
Numerade Educator
06:12

Problem 65

The velocity of a heavy meteorite entering the earth's atmosphere is inversely proportional to $\sqrt{s}$ when it is $s$ kilometers from the earth's center. Show that the meteorite's acceleration is inversely proportional to $s^{2}$.

Andrija Isakov
Andrija Isakov
Numerade Educator
01:54

Problem 66

A particle moves along the $x$ -axis with velocity $d x / d t=f(x) .$ Show that the particle's acceleration is $f(x) f^{\prime}(x)$.

Melissa Munoz
Melissa Munoz
Numerade Educator
06:00

Problem 67

For oscillations of small amplitude (short swings), we may safely model the relationship between the period $T$ and the length $L$ of a simple pendulum with the equation
$$T=2 \pi \sqrt{\frac{L}{g}}$$
where $g$ is the constant acceleration of gravity at the pendulum's location. If we measure $g$ in centimeters per second squared, we measure $L$ in centimeters and $T$ in seconds. If the pendulum is made of metal, its length will vary with temperature, either increasing or decreasing at a rate that is roughly proportional to $L$ In symbols, with $u$ being temperature and $k$ the proportionality constant,
$$\frac{d L}{d u}=k L.$$
Assuming this to be the case, show that the rate at which the period changes with respect to temperature is $k T / 2$.

Robert Daugherty
Robert Daugherty
Numerade Educator
01:40

Problem 68

Suppose that $f(x)=x^{2}$ and $g(x)=|x| .$ Then the composites
$$(f \circ g)(x)=|x|^{2}=x^{2} \quad \text { and } \quad(g \circ f)(x)=\left|x^{2}\right|=x^{2}$$
are both differentiable at $x=0$ even though $g$ itself is not differentiable at $x=0 .$ Does this contradict the Chain Rule? Explain.

Gregory Higby
Gregory Higby
Numerade Educator
04:34

Problem 69

Suppose that $u=g(x)$ is differentiable at $x=1$ and that $y=f(u)$ is differentiable at $u=g(1) .$ If the graph of $y=f(g(x))$ has a horizontal tangent at $x=1,$ can we conclude anything about the tangent to the graph of $g$ at $x=1$ or the tangent to the graph of $f$ at $u=g(1) ?$ Give reasons for your answer.

Andrija Isakov
Andrija Isakov
Numerade Educator
01:06

Problem 70

$\frac{d}{d x}(\sin x)=\cos x,$ if $x$ is measured in degrees or radians. Justify your answer.

Carson Merrill
Carson Merrill
Numerade Educator
01:27

Problem 71

The slope of the normal line to the curve $x=3 \cos t, y=3 \sin t$ at $t=\pi / 4$ is $-1 .$ Justify your answer.

Carson Merrill
Carson Merrill
Numerade Educator
03:30

Problem 72

Which of the following is $d y / d x$ if $y=\tan (4 x) ?$
(A) $4 \sec (4 x) \tan (4 x)$
(B) $\sec (4 x) \tan (4 x)$
(C) $4 \cot (4 x)$
(D) $\sec ^{2}(4 x)$
(E) $4 \sec ^{2}(4 x)$

Andrija Isakov
Andrija Isakov
Numerade Educator
01:39

Problem 73

Which of the following is $d y / d x$ if $y=\cos ^{2}\left(x^{3}+x^{2}\right) ?$
(A) $-2\left(3 x^{2}+2 x\right)$
(B) $-\left(3 x^{2}+2 x\right) \cos \left(x^{3}+x^{2}\right) \sin \left(x^{3}+x^{2}\right)$
(C) $-2\left(3 x^{2}+2 x\right) \cos \left(x^{3}+x^{2}\right) \sin \left(x^{3}+x^{2}\right)$
(D) $2\left(3 x^{2}+2 x\right) \cos \left(x^{3}+x^{2}\right) \sin \left(x^{3}+x^{2}\right)$
(E) $2\left(3 x^{2}+2 x\right)$

Adrian Co
Adrian Co
Numerade Educator
05:50

Problem 74

Use the curve defined by the parametric equations $x=t-\cos t, y=-1+\sin t$.

Which of the following is an equation of the tangent line to the curve at $t=0 ?$
(A) $y=x$
(B) $y=-x$
(C) $y=x+2$
(D) $y=x-2$
(E) $y=-x-2$

Andrija Isakov
Andrija Isakov
Numerade Educator
04:45

Problem 75

Use the curve defined by the parametric equations $x=t-\cos t, y=-1+\sin t$.

At which of the following values of $t$ is $d y / d x=0 ?$
(A) $t=\pi / 4$
(B) $t=\pi / 2$
(C) $t=3 \pi / 4$
(D) $t=\pi$
(E) $t=2 \pi$

Andrija Isakov
Andrija Isakov
Numerade Educator
01:18

Problem 76

Graph the function $y=2 \cos 2 x$ for $-2 \leq x \leq 3.5 .$ Then, on the same screen, graph
$$y=\frac{\sin 2(x+h)-\sin 2 x}{h}$$
for $h=1.0,0.5,$ and $0.2 .$ Experiment with other values of $h$ including negative values. What do you see happening as $h \rightarrow 0 ?$ Explain this behavior.

Carson Merrill
Carson Merrill
Numerade Educator
01:08

Problem 77

Graph $y=-2 x \sin \left(x^{2}\right)$ for $-2 \leq x \leq 3 .$ Then, on the same screen, graph
$$y=\frac{\cos \left[(x+h)^{2}\right]-\cos (x)^{2}}{h}$$
for $h=1.0,0.7,$ and $0.3 .$ Experiment with other values of $h$ What do you see happening as $h \rightarrow 0 ?$ Explain this behavior.

Carson Merrill
Carson Merrill
Numerade Educator
01:10

Problem 78

Let $u$ be a differentiable function of $x$.
(a) Show that $\frac{d}{d x}|u|=u^{\prime} \frac{u}{|u|}$.
(b) Use part (a) to find the derivatives of $f(x)=\left|x^{2}-9\right|$ and $g(x)=|x| \sin x$.

Carson Merrill
Carson Merrill
Numerade Educator
01:48

Problem 79

The geometric mean of $u$ and $v$ is $G=\sqrt{u v}$ and the arithmetic mean is $A=(u+v) / 2$ Show that if $u=x, v=x+c, c$ a real number, then
$$\frac{d G}{d x}=\frac{A}{G}.$$

Linh Vu
Linh Vu
Numerade Educator