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Chapter 3

Differentiation Rules

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Problem 1

Write the composite function in the form $ f(g(x)). $ [Identify the inner function $ u = g(x) $ and the outer function $ y = f(u). $ ] Then find the derivative $ dy/ dx. $
$ y = \sqrt[3]{1 + 4x} $

Heather Z.
Numerade Educator

Problem 2

Write the composite function in the form $ f(g(x)). $ [Identify the inner function $ u = g(x) $ and the outer function $ y = f(u). $ ] Then find the derivative $ dy/ dx. $
$ y = (2x^3 + 5)^4 $

Heather Z.
Numerade Educator

Problem 3

Write the composite function in the form $ f(g(x)). $ [Identify the inner function $ u = g(x) $ and the outer function $ y = f(u). $ ] Then find the derivative $ dy/ dx. $
$ y = \tan \pi x $

Heather Z.
Numerade Educator

Problem 4

Write the composite function in the form $ f(g(x)). $ [Identify the inner function $ u = g(x) $ and the outer function $ y = f(u). $ ] Then find the derivative $ dy/ dx. $
$ y = \sin(\cot x) $

Heather Z.
Numerade Educator

Problem 5

Write the composite function in the form $ f(g(x)). $ [Identify the inner function $ u = g(x) $ and the outer function $ y = f(u). $ ] Then find the derivative $ dy/ dx. $
$ y = e^{\sqrt{x}} $

Heather Z.
Numerade Educator

Problem 6

Write the composite function in the form $ f(g(x)). $ [Identify the inner function $ u = g(x) $ and the outer function $ y = f(u). $ ] Then find the derivative $ dy/ dx. $
$ y = \sqrt{2 - e^x} $

Heather Z.
Numerade Educator

Problem 7

Find the derivative of the function.
$ F(x) = (5x^6 + 2x^3)^4 $

dd
Deepak D.
Numerade Educator

Problem 8

Find the derivative of the function.
$ F(x) = (1 + x + x^2)^{99} $

PC
Partha Sarathi C.
Numerade Educator

Problem 9

Find the derivative of the function.
$ f(x) = \sqrt{5x + 1} $

Heather Z.
Numerade Educator

Problem 10

Find the derivative of the function.
$ f(x) = \frac {1}{\sqrt [3]{x^2 - 1}} $

Heather Z.
Numerade Educator

Problem 11

Find the derivative of the function.
$ f(\theta) = \cos (\theta^2) $

Heather Z.
Numerade Educator

Problem 12

Find the derivative of the function.
$ g(\theta) = \cos^2 \theta $

Heather Z.
Numerade Educator

Problem 13

Find the derivative of the function.
$ y = x^2 e^{-3x} $

Heather Z.
Numerade Educator

Problem 14

Find the derivative of the function.
$ f(t) = t \sin \pi t $

Heather Z.
Numerade Educator

Problem 15

Find the derivative of the function.
$ f(t) = e^{at} \sin bt $

Heather Z.
Numerade Educator

Problem 16

Find the derivative of the function.
$ g(x) = e^{x^2 - x} $

Heather Z.
Numerade Educator

Problem 17

Find the derivative of the function.
$ f(x) = (2x - 3)^4 (x^2 + x + 1)^5 $

Heather Z.
Numerade Educator

Problem 18

Find the derivative of the function.
$ g(x) = (x^2 + 1)^3 (x^2 + 2)^6 $

Heather Z.
Numerade Educator

Problem 19

Find the derivative of the function.
$ h(t) = (t +1)^{2/3} (2t^2 - 1)^3 $

Heather Z.
Numerade Educator

Problem 20

Find the derivative of the function.
$ F(t) = (3t - 1)^4 (2t + 1)^{-3} $

Heather Z.
Numerade Educator

Problem 21

Find the derivative of the function.
$ y = \sqrt \frac {x}{x + 1} $

Heather Z.
Numerade Educator

Problem 22

Find the derivative of the function.
$ y = (x + \frac {1}{x})^5 $

Heather Z.
Numerade Educator

Problem 23

Find the derivative of the function.
$ y = e^{\tan \theta} $

Heather Z.
Numerade Educator

Problem 24

Find the derivative of the function.
$ f(t) = 2^{t^3} $

Heather Z.
Numerade Educator

Problem 25

Find the derivative of the function.
$ g(u) = ( \frac {u^3 - 1}{u^3 +1})^8 $

Heather Z.
Numerade Educator

Problem 26

Find the derivative of the function.
$ s(t) = \sqrt \frac {1 + \sin t}{1 + \cos t} $

Heather Z.
Numerade Educator

Problem 27

Find the derivative of the function.
$ r(t) = 10^{2 \sqrt {t}} $

AK
Ayush K.
Numerade Educator

Problem 28

Find the derivative of the function.
$ f(z) = e^{z/(z - 1)} $

Heather Z.
Numerade Educator

Problem 29

Find the derivative of the function.
$ H(r) = \frac {(r^2 - 1)^3}{(2r + 1)^5} $

Heather Z.
Numerade Educator

Problem 30

Find the derivative of the function.
$ J(\theta) = \tan^2 (n \theta) $

Heather Z.
Numerade Educator

Problem 31

Find the derivative of the function.
$ F(t) = e^{t \sin 2t} $

Heather Z.
Numerade Educator

Problem 32

Find the derivative of the function.
$ F(t) = \frac {t^2}{\sqrt {t^3 + 1}} $

Heather Z.
Numerade Educator

Problem 33

Find the derivative of the function.
$ G(x) = 4^{C/x} $

Heather Z.
Numerade Educator

Problem 34

Find the derivative of the function.
$ U(y) = (\frac {y^4 + 1}{y^2 + 1})^5 $

Heather Z.
Numerade Educator

Problem 35

Find the derivative of the function.
$ y = \cos (\frac {1 - e^{2x}}{1 + e^{2x}}) $

Heather Z.
Numerade Educator

Problem 36

Find the derivative of the function.
$ y = x^2 e^{-1/x} $

Heather Z.
Numerade Educator

Problem 37

Find the derivative of the function.
$ y = \cot^2 (\sin \theta) $

Heather Z.
Numerade Educator

Problem 38

Find the derivative of the function.
$ y = \sqrt {1 + xe^{-2x}} $

Heather Z.
Numerade Educator

Problem 39

Find the derivative of the function.
$ f(t) = \tan (\sec(\cos t)) $

Heather Z.
Numerade Educator

Problem 40

Find the derivative of the function.
$ y = e^{\sin 2x} + \sin (e^{2x}) $

Heather Z.
Numerade Educator

Problem 41

Find the derivative of the function.
$ f(t) = \sin^2 (e^{\sin^2 t}) $

Heather Z.
Numerade Educator

Problem 42

Find the derivative of the function.
$ y = \sqrt {x + \sqrt {x + \sqrt {x}}} $

Heather Z.
Numerade Educator

Problem 43

Find the derivative of the function.
$ g(x) = (2ra^{rx} + n)^P $

Heather Z.
Numerade Educator

Problem 44

Find the derivative of the function.
$ y = 2^{3^{4^{x}}} $

Heather Z.
Numerade Educator

Problem 45

Find the derivative of the function.
$ y = \cos \sqrt {\sin (\tan \pi x)} $

Heather Z.
Numerade Educator

Problem 46

Find the derivative of the function.
$ y = [x + (x + \sin^2 x)^3]^4 $

Heather Z.
Numerade Educator

Problem 47

Find $y^{\prime}$ and $y^{\prime \prime}$
$$
y=\cos (\sin 3 \theta)
$$

Heather Z.
Numerade Educator

Problem 48

Find $ y' $ and $ y". $
$ y = \frac {1}{(1 + \tan x)^2} $

Frank L.
Numerade Educator

Problem 49

Find $y^{\prime}$ and $y^{\prime \prime}$
$$
y=\sqrt{1-\sec t}
$$

Frank L.
Numerade Educator

Problem 50

Find $ y' $ and $ y". $
$ y = e^{e^x} $

Heather Z.
Numerade Educator

Problem 51

Find an equation of the tangent line to the curve at the given point.
$ y = 2^x, (0, 1) $

Heather Z.
Numerade Educator

Problem 52

Find an equation of the tangent line to the curve at the given point.
$ y = \sqrt {1 + x^3}, (2, 3) $

Heather Z.
Numerade Educator

Problem 53

Find an equation of the tangent line to the curve at the given point.
$ y = \sin (\sin x), (\pi, 0) $

Heather Z.
Numerade Educator

Problem 54

Find an equation of the tangent line to the curve at the given point.
$ y = xe^{-x^2}, (0, 0) $

Heather Z.
Numerade Educator

Problem 55

(a) Find an equation of the tangent line to the curve $ y = 2/(1 + e^{-x}) $ at the point (0, 1).
(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

Heather Z.
Numerade Educator

Problem 56

(a) The curve $ y = \mid x \mid /\sqrt {2 - x^2} $ is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point (1, 1).
(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

Heather Z.
Numerade Educator

Problem 57

(a) If $ f(x) = x \sqrt {2 - x^2}, $ find $ f'(x). $
(b) Check to see that your answer to part (a) is reasonable by comparing the graph of $ f $ and $ f' $.

Heather Z.
Numerade Educator

Problem 58

The function $ f(x) = \sin (x + \sin 2x), 0 \le x \le \pi, $ arises in applications to frequency modulation (FM) synthesis.
(a) Use a graph of $ f $ produced by a calculator lo make a rough sketch of the graph of $ f'. $
(b) Calculate $ f'(x) $ and use this expression, with a calculator, to graph $ f'. $ Compare with your sketch in part (a).

Heather Z.
Numerade Educator

Problem 59

Find all points on the graph of the function $ f(x) = 2 \sin x + \sin^2 x $ at which the tangent line is horizontal.

Heather Z.
Numerade Educator

Problem 60

At what point on the curve $ y = \sqrt {1 + 2x} $ is the tangent line perpendicular to the line $ 6x + 2y = 1? $

Heather Z.
Numerade Educator

Problem 61

If $ F(x) = f(g(x)), $ where $ f(-2) = 8, f'(-2) =4, f'(5) = 3, g(5) = -2, $ and $ g'(5) = 6, $ find $ F'(5). $

Heather Z.
Numerade Educator

Problem 62

If $ h(x) = \sqrt {4 + 3f(x)}, $ where $ f(1) = 7 $ and $ f'(1) = 4, $ find $ h'(1). $

Heather Z.
Numerade Educator

Problem 63

A table of values for $ f, g, f' , $ and $ g' $ is given.
(a) If $ h(x) = f(g(x)), $ find $ h'(1). $
(b) If $ H(x) = g(g(x)), $ find $ H(1). $

Heather Z.
Numerade Educator

Problem 64

Let $ f $ and $ g $ be the function in Exercise 63.
(a) If $ F(x) = f(f(x)), $ find $ F'(2). $
(b) If $ G(x) = g(g(x)), $ find $ G'(3). $

Heather Z.
Numerade Educator

Problem 65

If $ f $ and $ g $ are the functions whose graphs are shown, let $ u(x) = f(g(x)), v(x) = g(f(x)), $ and $ w(x) = g(g(x)). $ Find each derivative, if it exists. If it does not exist, explain why.
(a) $ u'(1) $
(b) $ v'(1) $
(c) $ w'(1) $

Carson M.
Numerade Educator

Problem 66

If $ f $ is the function whose graph is shown, let $ h(x) = f(f(x)) $ and $ g(x) = f(x^2). $ Use the graph of $ f $ to estimate the value of each derivative,
(a) $ h'(2) $
(b) $ g'(2) $

Heather Z.
Numerade Educator

Problem 67

If $ g(x) = \sqrt {f(x)}, where Ihe graph of $ f $ is shown, evaluate $ g'(3). $

Heather Z.
Numerade Educator

Problem 68

Suppose $ f $ differentiable on $ \mathbb{R} $ and $ \alpha $ is a real number. Let $ F(x) = f(x^{\alpha}) $ and $ G(x) = [f(x)]^{\alpha}. $ Find expressions for
(a) $ F'(x) $ and (b) $ G'(x). $

Heather Z.
Numerade Educator

Problem 69

Suppose $ f $ is differentiable on $ \mathbb{R}. $ Let $ F(x) = f(e^x) $ and $ G(x) = e^{f(x)}. $ Find expressions for
(a) $ F'(x) $ (b) $ G'(x). $

Heather Z.
Numerade Educator

Problem 70

Let $ g(x) = e^{\alpha} + f(x) $ and $ h(x) = e^{kx}, $ where $ f(0) = 3, f'(0) = 5, $ and $ f" (0) = -2. $
(a) Find $ g'(0) $ and $ g"(0) $ in terms of $ c $.
b) In terms of $ k, $ find an equation of the tangent line to the graph of $ h $ at the point where $ x = 0. $

Heather Z.
Numerade Educator

Problem 71

Let $ r(x) = f (g(h(x))), $ where $ h(1) = 2, g(2) = 3, h'(1) = 4, g'(2) = 5, $ and $ f'(3) = 6. $ Find $ r'(1). $

Heather Z.
Numerade Educator

Problem 72

If $ g $ is a twice differentiable function and $ f(x) = xg(x^2), $ find $ f" $ in terms of $ g, g', $ and $ g". $

Heather Z.
Numerade Educator

Problem 73

If $ F(x) = f(3f(4f(x))), $ where $ f(0) = 0 $ and $ f' (0) = 2, $ find $ F' (0). $

Heather Z.
Numerade Educator

Problem 74

If $ F(x) = f(xf(xf(x))), $ where $ f(1) = 2, f'(2) = 3, f'(1) = 4, f'(2) = 5, $ and $ f'(3) = 6, $ find $ F'(1). $

Heather Z.
Numerade Educator

Problem 75

Show that the function $ y = e^{2x} (A \cos 3x + B \sin 3x) $ satisfies the differential equation $ y" - 4y' + 13y = 0. $

AK
Anjali K.
Numerade Educator

Problem 76

For what values of $ r $ does the function $ y = e^{rx} $ satisfy the differential equation $ y" - 4y' + y = 0? $

Heather Z.
Numerade Educator

Problem 77

Find the 50th derivative of $ y = \cos 2x. $

Heather Z.
Numerade Educator

Problem 78

Find the 1000th derivative of $ f(x) = xe^{-x}. $

Heather Z.
Numerade Educator

Problem 79

The displacement of a particle on a vibrating string is given by the equation $ s(t) = 10 + \frac {1}{4} \sin (10 \pi t) $ where $ s $ is measured in centimeters and $ t $ in seconds. Find the velocity of the particle after $ t $ seconds.

Gregory H.
Numerade Educator

Problem 80

If the equation of motion of a particle is given by $ s = A \cos (wt + 8), $ the particle is said to undergo simple harmonic motion.
(a) Find the velocity of the particle at time $ t. $
(b) When is the velocity 0?

Heather Z.
Numerade Educator

Problem 81

A Cepheid variable star is a star whose brightness alternately increases and decreases. The most easily visible such star is Delta Cephei, for which the interval between times of maximum brightness is 5.4 days. The average brightness of this star is 4.0 and its brightness changes by $ \pm O.35. $ In view of these data, the brightness of Delta Cephei at time $ t, $ where $ t $ is measured in days, has been modeled by the function
$ B(t) = 4.0 + 0.35 \sin (\frac {2 \pi t}{5.4}) $
(a) Find the rate of change of the brightness after $ t $ days.
(b) Find, correct to two decimal places, the rate of increase alter one day.

Tyler M.
Numerade Educator

Problem 82

In Example 1.3.4 we arrived at a model for the length of daylight (in hours) in Philadelphia on the $ t $ th day of the year:
$ L(t) = 12 + 2.8 \sin [ \frac {2 \pi}{365}(t - 80] $
Use this model to compare how the number of hours of daylight is increasing in Philadelphia on March 21 and May 21.

Frank L.
Numerade Educator

Problem 83

The motion of a spring that is subject to a fictional force or a damping force (such as a shock absorber in a car) is often modeled by the product of an exponential function and a sine or cosine function. Suppose the equation of motion of a point on such a spring is
$ s(t) = 2e^{-1.5t} \sin 2 \pi t $
where $ s $ is measured in centimeters and $ t $ in seconds. Find the velocity after $ t $ seconds and graph both the position and velocity functions for $ 0 \le t \le 2. $

Heather Z.
Numerade Educator

Problem 84

Under certain circumstances a rumor spreads according to the equation
$$
p(t)=\frac{1}{1+a e^{-k t}}
$$
where $p(t)$ is the proportion of the population that has heard the rumor at time $t$ and $a$ and $k$ are positive constants. [In Section 9.4 we will see that this is a reasonable equation for $p(t) .]$
(a) Find $\lim _{t \rightarrow \infty} p(t)$
(b) Find the rate of spread of the rumor.
(c) Graph $p$ for the case $a=10, k=0.5$ with $t$ measured in hours. Use the graph to estimate how long it will take for $80 \%$ of the population to hear the rumor.

Heather Z.
Numerade Educator

Problem 85

The average blood alcohol concentration (BAC) of eight male subjects was measured after consumption of 15 mL of ethanol (corresponding to one alcoholic drink). The resulting data were modeled by the concentration function
$ C(t) = 0.0225te^{0.0467t} $
where $ t $ is measured in minutes after consumption and C is measured in mg/mL.
(a) How rapidly was the BAC increasing alter 10 minutes?
(b) How rapidly was it decreasing half an hour later?

Tyler M.
Numerade Educator

Problem 86

In Section 1.4 we modeled the world population from 1900 to 2010 with the exponential function
$$
P(t)=(1436.53) \cdot(1.01395)^{t}
$$
where $t=0$ corresponds to the year 1900 and $P(t)$ is measured in millions. According to this model, what was the rate of increase of world population in $1920 ?$ In $1950 ?$ In $2000 ?$

Tyler M.
Numerade Educator

Problem 87

A panicle moves along a straight line with displacement $ s(t), $ velocity $ v(t), $ and acceleration $ a(t). $ Show that
$ a(t) = v(t) \frac {dv}{ds} $
Explain the difference between the meanings of the derivatives $ dv/dt $ and $ dv/ds. $

Heather Z.
Numerade Educator

Problem 88

Air is being pumped into a spherical weather balloon. At any time $ t, $ the volume of the balloon is $ V(t) $ and its radius is $ r(t). $
(a) What do the derivatives $ dV/dr $ and $ dV/dt $ represent?
(b) Express $ dV/dt $ in terms of $ dr/dt. $

Heather Z.
Numerade Educator

Problem 89

The flash unit on a camera operates by storing charge on a capacitor and releasing it suddenly when the flash is set off. The following data describe the charge $ Q $ remaining on the capacitor (measured in microcoulombs, $ \mu C $ ) at time $ t $ (measured in seconds).
(a) Use a graphing calculator or computer to find an exponential model for the charge.
(b) The derivative $ Q'(t) $ represents the electric current (measured in microamperes, $ \mu A $ ) flowing from the capacitor to the flash bulb. Use part (a) to estimate the current when $ t = 0.04 s. $ Compare with the result of Example 2.1.2.

Heather Z.
Numerade Educator

Problem 90

The table gives the US population from 1790 to 1860.
(a) Use a graphing calculator or computer to fit an exponential function to the data. Graph the data points and the exponential model. How good is the fit?
(b) Estimate the rates of population growth in 1800 and 1850 by averaging slopes of secant lines.
(c) Use (he exponential model in part (a) to estimate the rates of growth in 1800 and 1850. Compare these estimates with the ones in part (b).
(d) Use the exponential model to predict the population in 1870. Compare with the actual population of 38,358,000. Can you explain the discrepancy?

Heather Z.
Numerade Educator

Problem 91

Computer algebra systems have commands that differentiate functions, but the form of the answer may not be convenient and so further commands may be necessary to simplify the answer.
(a) Use a CAS to find the derivative in Example 3 and compare with the answer in that example. Then use the simplify command and compare again.
(b) Use a CAS to find the derivative in Example 6. What happens if you use the simplify command? What happens if you use the factor command? Which form of the answer would be best for locating horizontal tangents?

Frank L.
Numerade Educator

Problem 92

(a) Use a CAS lo differentiate the function
$ f(x) = \sqrt \frac {x^4 - x +1}{x^4 + x +1} $
and to simplify the result.
(b) Where does the graph of $ f $ have horizontal tangents?
(c) Graph $ f $ and $ f' $ on the same screen. Are the graphs consistent with your answer to part (b)?

Frank L.
Numerade Educator

Problem 93

Use the Chain Rule lo prove the following.
(a) The derivative of an even (unction is an odd function.
(b) The derivative of an odd (unction is an even function.

Heather Z.
Numerade Educator

Problem 94

Use the Chain Rule and the Product Rule to give an alternative proof of the Quotient Rule.
[ $ Hint: $ Write $ f(x)/ g(x) = f(x)[g(x)]^{-1}.] $

Heather Z.
Numerade Educator

Problem 95

(a) If $ n $ is a positive integer, prove that
$ \frac {d}{dx} (\sin^a x \cos nx) = n \sin^{a-1} x \cos (n + 1)x $
(b) Find a formula for the derivative of $ y = \cos^a x \cos nx $ that is similar to the one in part (a).

Heather Z.
Numerade Educator

Problem 96

Suppose $ y = f(x) $ is a curve that always lies above the $ x $-axis and never has a horizontal tangent, where $ f $ is differentiable everywhere. For what value of $ y $ is the rate of change of $ y^5 $ with respect to $ x $ eighty times the rate of change of $ y $ with respect to $ x? $

Heather Z.
Numerade Educator

Problem 97

Use the Chain Rule lo show that if $ \theta $ is measured in degrees, then
$ \frac {d}{d \theta} (\sin \theta) = \frac {\pi}{180} \cos \pi $
(This gives one reason for the convention that radian measure is always used when dealing with trigonometric functions in calculus: the differentiation formulas would not be as simple if we used degree measure.)

Heather Z.
Numerade Educator

Problem 98

(a) Write $ \mid x \mid = \sqrt {x^2} $ and use the Chain Rule to show that
$ \frac {d}{dx} \mid x \mid = \frac {x}{\mid x \mid} $
(b) If $ f(x) = \mid \sin x \mid, $ find $ f'(x) $ and sketch the graph of $ f $ and $ f'. $ Where is $ f $ is not differentiable?
(c) If $ g(x) = \sin \mid x \mid, $ find $ g'(x) $ and sketch the graphs of $ g $ and $ g'. $ Where is $ g $ not differentiable?

Heather Z.
Numerade Educator

Problem 99

If $ = f(u) $ and $ u = g(x), $ where $ f $ and $ g $ are twice differentiable functions, show that
$ \frac {d^2 y}{dx^2} = \frac {d^2 y}{du^2} (\frac {du}{dx})^2 + \frac {dy}{du} \frac {d^2 u}{dx^2} $

Heather Z.
Numerade Educator

Problem 100

If $ y = f(u) $ and $ u = g(x), $ where $ f $ and $ g $ possess third derivatives, find a formula for $ d^3 y/dx^3 $ similar to the one given in Exercise 99.

Heather Z.
Numerade Educator