3.1 Deﬁnition of the Derivativedeturck/m103/rogawski/chap3...3.1 Deﬁnition of the Derivative 3 In Exercises 6–9, compute the derivative of the quadratic polynomial at the point

CHAPTER

3 Differentiation3.1 Definition of the Derivative

Preliminary Questions

1. What are the two ways of writing the difference quotient?

2. Explain in words what the difference quotient represents.

In Questions 3–5, f (x) is an arbitrary function.

3. What does the following quantity represent in terms of the graph of f (x)?

f (8) − f (3)8 − 3

4. For which value of x is

f (x) − f (3)x − 3 =

f (7) − f (3)4

?

5. For which value of h is

f (2 + h) − f (2)h

= f (4) − f (2)4 − 2 ?

6. To which derivative is the quantity

tan( π4 + .00001) − 1.00001

a good approximation?

7. What is the equation of the tangent line to the graph at x = 3 of a function f (x) such thatf (3) = 5 and f ′(3) = 2?

In Questions 8–10, let f (x) = x2.

1

2 Chapter 3 Differentiation

8. The expression

f (7) − f (5)7 − 5

is the slope of the secant line through two points P and Q on the graph of f (x). What arethe coordinates of P and Q?

9. For which value of h is the expression

f (5 + h) − f (5)h

equal to the slope of the secant line between the points P and Q in Question 8?

10. For which value of h is the expression

f (3 + h) − f (3)h

equal to the slope of the secant line between the points (3, 9) and (5, 25) on the graph off (x)?

Exercises

1. Which of the following lines is tangent to the curve?

A

B C

D

Figure 1

Lines A and D are tangent to the curve.

In Exercises 2–5, use the limit definition to find the derivative of the following linear functions(for any a; the derivative does not depend on a).

2.f (x) = 3x − 2

3. f (x) = 2limh→0

f (a + h) − f (a)h

= limh→0

2 − 2h

= limh→0

0 = 0.4.

g(t) = 9 − t5. k(z) = 16z + 9

limh→0

k(a + h) − k(a)h

= limh→0

16(a − h) + 9 − (16a + 9)h

= limh→0

16h

h= 0.

3.1 Definition of the Derivative 3

In Exercises 6–9, compute the derivative of the quadratic polynomial at the point indicatedusing both forms of the derivative definition.

6.x2 + 9x ; a = 0

7. x2 + 9x ; a = 2Let f (x) = x2 + 9x . Thenf ′(2) = lim

h→0f (2 + h) − f (2)

h= lim

h→0(2 + h)2 + 9(2 + h) − 22

h= lim

h→013h + h2

h= limh→0

(13 + h) = 13.8.

3x2 + 4x + 2; a = −19. 9 − 3x2; a = 0

Let f (x) = 9 − 3x2. Then f ′(0) = limh→0

f (0 + h) − f (0)h

= limh→0

9 − 3h2 − 9h

= limh→0

−3h2h

= limh→0

(−3h) = 0.

10.(Exercises 10–11 adapted from The Logic of Teaching Calculus, Susanna S. Epp.) Draw thetangent lines to the graph of f (x) = x2 − 6x + 7 at the points (1, 2), (2, −1), and (5, 2).State whether the derivative is positive or negative or zero at these points.

Figure 2 Graph of f (x) = x2 − 6x + 7.

11. R & W Let f be the function shown in Figure 3.(a) Given a number h, describe in words what the quantity

f (2 + h) − f (2)h

represents in terms of the graph.(b) Use the graph to estimate this quantity for h = .5 and h = −.5.(c) What does f ′(2) represent? Is it larger or smaller than the two numbers computed in

(b)? Explain your reasoning.

0.5 1 1.5 2 32.5

5

4

3

2

1

Figure 3

(a) The difference quotient

f (2 + h) − f (2)h

represents the slope of the secant line through the points (2, f (2)) and(2 + h, f (2 + h)) on the curve.

(b) For h = −.5, this quantity is 4−2−.5 = −4. For h = .5, this quantity is approximately1.2−2

.5 = −1.6.(c) The derivative f ′(2) represents the slope of the tangent line to the curve at x = 2. It is

between the two quantities computed in (b).


12.

GU Let f (x) = 2x2 − 3x − 5.(a) Show that for h �= 0, the slope of the secant line between the points (2, f (2)) and

(2 + h, f (2 + h)) is equal to 2h + 5.(b) Use this formula to compute the slope of the secant line through the points P and Q on

the graph where x = 2 and x = 2.5.(c) Sketch a graph of f . Indicate the points P and Q and the secant line passing through

them.

13. Let f (x) = √x . Let P = (2, f (2)) and Q = (2 + h, f (2 + h))(a) Show that for h �= 0, the slope of the secant line between the points P and Q is equal

to 1/(√

2 + h + √2).(b) Use (a) to calculate the slope of the secant line through P and Q for

h = .1, .01, −.1, −.01.(c) Describe (in words and a graph) the limiting position of these secant lines as h → 0.(d) Evaluate f ′(2) using the formula in (a).

(a) The slope of the secant line between points P = (2, f (2)) and Q = (2 + h, f (2 + h))is equal to

f (2 + h) − f (2)(2 + h) − 2 =

√2 + h − √2

h·√

2 + h + √2√2 + h + √2 = 1/

√2 + h + √2.

(b)h −.1 −.01 .01 .1

1/(√

2 + h + √2) .358 .354 .353 .349(c) The limiting position of the secant lines is the tangent line at x = 2.(d) f ′(2) = lim

h→0f (2 + h) − f (2)

h= lim

h→01√

2 + h + √2 =1

2√

2≈ .354.

14.

Let f be the function whose graph is shown in Figure 4.(a) Use the graph to estimate

f (2.5) − f (1)2.5 − 1 .

(b) What does this quantity represent?

Figure 4

15. Describe what each of the following quantities represents in terms of the graph off (x) = sin x .(a) f ′( π4 )

(b)sin 1.3 − sin .9

.4(c) sin 3 − sin 1Consider the graph of y = sin x .(a) The quantity sin′( π4 ) represents the slope of the tangent line to the graph at a = π4 .(b) The quantity

sin 1.3 − sin .9.4

represents the slope of the secant line to the graph

between the points (.9, sin .9) and (1.3, sin 1.3).(c) The quantity sin 3 − sin 1 represents the numerator of the slope of the secant line to the

graph between the points (1, sin 1) and (3, sin 3).

In Exercises 16–35, compute the derivative of the function at the point indicated using thedefinition as a limit of difference quotients and find the equation of the tangent line.

16.3x2 + 2x , a = 2

17. 3x2 + 2x , a = −1Let f (x) = 3x2 + 2x . Then

f ′(−1) = limh→0

f (−1 + h) − f (−1)h

= limh→0 3(1 − 2h + h2) − 2 + 2h − 1h

= limh→0

3(1 − 2h + h2) − 2 + 2hh

= limh→0

−4h + 3h2h

= −4

The tangent line at a = −1 is y = f ′(−1)(x + 1) + f (−1) = −4(x + 1) + 1 = −4x − 3.


18.x3, a = 2

19. x3, a = 3Let f (x) = x3. Then

limh→0

f (3 + h) − f (3)h

= limh→0

27 − 27h + 9h2 + h3 − 27h

= limh→0

27 + 9h + h2 = 27.

At a = 3, the tangent line is y = f ′(3)(x − 3) + f (3) = 27(x − 3) + 27 = 27x − 54.20.x3 + x , a = 0

21. 3t3 + 2t , a = 4Let g(t) = 3t3 + 2t . Then

g′(4) = limh→0

g(4 + h) − g(4)h

= limh→0

3(4 + h)3 + 2(4 + h) − 200h

= limh→0

3h3 + 36h2 + 146hh

= limh→0

(3h2 + 36h + 146) = 146.

The tangent line at a = 4 is y = g(4) + g′(4)(t − 4) = 200 + 146(t − 4), which simplifiesto y = 146t − 384.22.2x3 − x2, a = −1

23. x−1, a = 3Let f (x) = x−1. Then

f ′(3) = limh→0

f (3 + h) − f (3)h

= limh→0

(3 + h)−1 − 3−1h

= limh→0

13+h −

(13

)h

= limh→0

3−3−h3(3+h)

h

= limh→0

−h(9 + 3h)h = −

1

9

The tangent at a = 3 is y = f ′(3)(x − 3) + f (3) = ( 13

)− ( 19

)(x − 3).

24.x−2, a = 1

25. x−2, a = −1Let f (x) = x−2. Then

f ′(−1) = limh→0

f (−1 + h) − f (−1)h

= limh→0

11−2h+h2 − 1

h

= limh→0

1 − 1 + 2h − h2h(1 − 2h + h2 = 2

The tangent line at a = −1 is y = f ′(−1)(x + 1) + f (−1) = 2(x + 1) + 1 = 2x + 3.26.4

x, a = 2

27.2

1 − t , a = −1Let g(t) = 21−t . Then

g′(t) = limh→0

g(t + h) − g(t)h

= limh→0

21−(t+h) − 21−t

h

= limh→0

21−x−h − 21−t

h= lim

h→0

2−2t−2+2t+2h(1−t−h)(1−t)

h= 2

(1 − t)2Then f ′(−1) = 2

(1+1)2 = 12 . At a = −1, the tangent line isy = g′(t)(x + 1) + g(t) = ( 1

2

)(x + 1) + 1. = ( x

2

)+ 32


28. √x , a = 9

29.1√x

, a = 9

Let f (x) = √x . Then

f ′(9) = limh→0

f (9 + h) − f (9)h

= limh→0

1√(9+h) − 13

h

= limh→0

3−√(9+h)3√

(9+h) · 3+√

(9+h)3+√(9+h)

h= lim

h→0

9−9−h9√

(9+h)+3(9+h)h

= − 154

At a = 9 the tangent line is y = f ′(9)(x − 9) + f (9) = (− 154

)(x − 9) + 1

3= (− x

54

)+ ( 12

).

30.1

t + 9 , a = 231. x + 1x − 1 , a = 3

Let f (x) = x + 1x − 1 . Then

f ′(3) = limh→0

f (3 + h) − f (3)h

= limh→0

4+h2+h − 2

h

= limh→0

−h2+hh

= limh→0

−12 + h = −

1

2.

The tangent line at a = 3 is y = f (3) + f ′(3)(x − 3) = 2 − 12(x − 3) or y = 7

2− x

2.

32. zz − 1 , a = 233. t−3, a = 1Let g(t) = 1

t3. Then

g′(1) = limh→0

g(1 + h) − g(h)h

= limh→0

1(1+h)3 − 1

h

= limh→0

h(3+3h+h2)(1+h)3

h= lim

h→03 + 3h + h2

(1 + h)3 = −3.

The tangent line at a = 1 is y = g(1) + g′(1)(t − 1) = 1 − 3(t − 1) or y = 4 − 3t .34.

t4, a = 235. (x2 + 1)−1, a = 0

Let f (x) = 1x2 + 1 . Then

f ′(0) = limh→0

f (0 + h) − f (0)h

= limh→0

1(0+h)2+1 − 1

h

= limh→0

−h2h2+1h

= limh→0

−hh2 + 1 = 0.

The tangent line at 0 is y = f (0) + f ′(0)(x − 0) = 1 + 0(x − 1) or y = 1.36.

Find the x-coordinates of the points on the curve shown in Figure 5, where the tangent lineis horizontal.

Figure 5

37. Which of the curves in Figure 6 is the graph of a function with the property that f ′(x) > 0for all x? Which has the property that f ′(x) < 0?


A B C

D E

Figure 6

Curve D has f ′(x) > 0 for all x (its slope is always positive), whereas curve C hasf ′(x) < 0 for all x (its slope is always negative).38.For the graph in Figure 7, determine the intervals along the x-axis on which the derivativeis positive.

Figure 7

Each of the limits in Exercises 39–44 represents a derivative f ′(a). Find f and a.

39. limh→0

(5 + h)3 − 125h

limh→0

(5 + h)3 − 125h

represents f ′(a) where f (x) = x3 and a = 5.40.

limx→5

x3 − 125x − 541. lim

h→05(2+h) − 25

h

limh→0

5(2+h) − 25h

represents f ′(a) where f (x) = 5x and a = 2.42.

limh→0

5h − 1h43. lim

h→0sin( π

3+ h) − .5h

limh→0

sin( π6

+ h) − .5h

represents f ′(a) where f (x) = sin x and a = π6 .44.limx→ 14

x−1 − 4x − 1

445. Consider the “curve” y = 2x + 8 (a straight line!). What is the tangent line at the point(1, 10)? Describe the tangent line at an arbitrary point.

Since y = 2x + 8 represents a straight line, the tangent line at any point is the line itself,y = 2x + 8.46.Suppose that f is a function such that f (2 + h) − f (2) = 3h2 + 3h.

(a) What is the slope of the secant line through (2, f (2)) and (3, f (3))?(b) What is f ′(2)?

47. Let f be the function whose graph is shown in Figure 8.(a) Find the values of x such that f ′(x) = 0.(b) For which value of x is f ′(x) largest?(c) On which intervals is f ′(x) negative?

1 2 3 4 5 6 7 8.5

Figure 8


(a) f ′(x) = 0 at x ≈ − 13 , 13 , 1.4, 3 14 since the tangent lines are horizontal there.(b) f ′(x) is largest at x = 0 since the tangent line has the steepest slope there.(c) f ′(0) is negative on the (approximate) intervals (−1, − 1

3), ( 1

3, 1.4), (3 1

4, 8.2).

48.Calculate the difference quotient

f (1 + h) − f (1)h

for a few values of h and estimate f ′(1) where f (x) is a function taking on the followingvalues:

x 1.03 1.02 1.01 1f (x) 0.5148 0.5234 0.5319 0.5403

x .97 .98 .99f (x) 0.5653 0.557 0.5486

49. GU Let f (x) = cos x .(a) Estimate f ′( π

6) to at least three decimal places.

(b) Write an equation for an approximate tangent line at the point ( π6,

√3

2).

(c) Use a CAS to plot f and the approximate tangent line.

(a)

h −.01 −.001 .001 .01cos( π

6+ h) −

√3

2

h−.495662 −.499567 −.500433 −.504322

With f (x) = cos x , we estimate f ′( π6) ≈ −.500.

(b) The tangent line is y = f ( π6) + f ′( π

6)(x − π

6) ≈ .866 − .500(x − .524) or

y ≈ 1.128 − .500x .

(c)

1

0 150.

Let m denote the slope of the tangent line to the graph of f (x) = sin x at x = π2

.(a) Sketch the graph of f and make a guess for the value of m.(b) Calculate the slope of the secant line between x = π

2and x = π

2+ h for at least three

small positive and negative values of h. Are these calculations consistent with theguess made in (a)?

51. R & W Consider the slopes of the secant lines between x = 1 and x = 1 + h for each ofthe graphs in Figure 9. In each case, state whether f ′(1) is less than or greater than theslope of the secant line when h > 0. Same question for h < 0. Explain your reasoning byreferring to the graph.

1 1 1

(A) (C)(B)

Figure 9

(a) f ′(1) is less because it is decreasing at a faster rate.(b) f ′(1) is greater because it is increasing at a faster rate.(c) f ′(1) is less because it is increasing at a slower rate.

52.Apply the method of Example ?? to f (x) = sin x to determine f ′ ( π

4

)accurately to four

decimal places.53. R & W Let f (x) = x5/2.


(a) Use Figure 10 to explain why the following inequalities hold:

f ′(4) ≤ f (4 + h) − f (4)h

if h < 0

f ′(4) ≥ f (4 + h) − f (4)h

if h > 0

4

Figure 10 Graph of f (x) = x5/2.

(b) Estimate f ′(4) to four decimal places.

(a) The slope of the secant line between points (4, f (4)) and (4 + h, f (4 + h)) isf (4+h)− f (h)

h. x5/2 is a smooth curve increasing at a faster rate as x → ∞. Therefore, if

h > 0, then the slope of the secant line is greater than the slope of the tangent line atf (4), which happens to be f ′(4). Likewise, if h < 0, the slope of the secant line is lessthan the slope of the tangent line at f (4), which happens to be f ′(4).

(b) We know that f ′(4) = limh→0 f (4 + h) − f (4)h

= limh→0 (4 + h)5/2 − 32h

. Creating a

table with values of h close to zero:h −.0001 −.00001 .00001 .0001

(4 + h)5/2 − 32h

19.999625 19.99999 20.0000 20.0000375

f ′(4) ≈ 19.999.54.

R & W Apply the method of Example ?? to f (x) = cos x to determine f ′( π5) accurately

to four decimal places. Explain how the method works in this case (sketch a graph and referto it).

55. R & W Let f (x) = sec x .(a) Use Figure 11 to explain why the following inequalities hold:

f ′(1) ≤ f (1 + h) − f (1)h

if h < 0

f ′(1) ≥ f (1 + h) − f (1)h

if h > 0

(b) Estimate f ′(1) to four decimal places.

2π

2π–

Figure 11 Graph of f (x) = sec x .


(a) The slope of the secant line between points (1, f (1)) and (1 + h, f (1 + h)) isf (1+h)− f (h)

h. sec x between the values of 0 and π

2is a smooth curve increasing at a faster

rate as x → π2 . Therefore, if h > 0, then the slope of the secant line is greater than theslope of the tangent line at f (1) which happens to be f ′(1). Likewise, if h < 0, the theslope of the secant line is less than the slope of the tangent line at f (1), which happensto be f ′(1).

(b) We know that

f ′(1) = limh→0

f (1 + h) − f (1)h

= limh→0

sec(1 + h) − sec(1)h

.

Create a table with values of h close to zero:h −.001 −.0001 .0001 .001

sec(1 + h) − sec(1)h

2.87707 2.88193 2.887899 22.88193

f ′(1) ≈ 2.8819.56.

R & W The graph of f (x) = 2x is shown in Figure 12.(a) Referring to the graph, explain why the inequality

f ′(0) ≤ f (h) − f (0)h

holds for positive values of h, while the opposite inequality holds for h negative.(b) Use (a) with small values of h to show that .69314 ≤ f ′(0) ≤ .69315.(c) Use the same procedure to determine f ′(x) to four decimal places for x = 1, 2, 3, 4.(d) Now compute the ratios f ′(x)/ f ′(0) for x = 1, 2, 3, 4. Can you guess an

(approximate) formula f ′(x) for general x?

Figure 12 Graph of 2x .

57. Let f be the function whose graph is shown in Figure 13 and consider

f (2 + h) − f (2)h

as a function of h. Is this an increasing or a decreasing function of h? In other words, doesit get larger or smaller as h increases?

1 2 3 4 5

Figure 13

For the depicted graph, the difference quotientf (2 + h) − f (2)

his an increasing function

of h since the slopes of the secant lines between (2, f (2)) and a nearby point increase asthe nearby point moves from left to right.

Further Insights and Challenges58.

Symmetric Difference Quotient The symmetric difference quotient at x = a is de-fined for h �= 0 by

f (a + h) − f (a − h)2h

(1)

Explain how (1) can be interpreted as the slope of a secant line. Hint: consider the points onthe graph with x = a + h and x = a − h.

59. For small h, the symmetric difference quotient usually gives a better approximation to thederivative than the ordinary difference quotient. Let f (x) = 2x and a = 0. Computethe symmetric difference quotient with h = 0.001 and the ordinary difference quotientswith h = ±0.001. Compare with the actual value of f ′(0) which, to 8 decimal places, is0.69314718.

Let f (x) = 2x and a = 0.

3.2 The Derivative as a Function 11

The ordinary difference quotient for h = −.001 is .69290701 and for h = .001 is.69338746.The symmetric difference quotient for h = −.001 is .69314724 and for h = .001 is.69314724.Clearly the symmetric difference quotient gives a better estimate of the derivativef ′(0) ≈ .69314718.

60.Which of the two functions in Figure 14 satisfies the inequality

f (a + h) − f (a − h)2h

≤ f (a + h) − f (a)h

for h > 0? Explain in terms of secant lines.

Figure 14

3.2 The Derivative as a Function


1. What is the slope of the tangent line through the point (2, f (2)) if f is a function such thatf ′(x) = x3?

2. Suppose that f and g are differentiable functions such that f ′(1) = 3 and g′(1) = 5.Which of the following derivatives can be computed using the rules of this section and theinformation given? Evaluate the derivatives that can be computed.(a) ( f + g)′(1)(b) ( f − g)′(1)(c) ( f g)′(1)(d) (3 f + 2g)′(1)(e)

(f

g

)′(1)

3. Which of the following functions can be differentiated using the rules covered in this sec-tion? Explain.(a) x2

(b) 2x

(c)1√x

(d) x−4/5

(e) sin x(f) (x + 1)3

4. Which algebraic identity is used to prove the Power Rule for positive integer exponents?Explain how it is used.

Exercises

In Exercises 1–6, calculate f ′(x) using the limit definition.

1. 4x − 3Let f (x) = 4x − 3 Then,

f ′(x) = limh→0

f (x + h) − f (x)h

= limh→0

4(x + h) − 3 − (4x − 3)h

= limh→0

4h

h= 4.


2.x2 + x

3. 1 − 2x2

f ′(x) = limh→0 f (x + h) − f (x)h

= limh→0 1 − 2(x + h)2 − (1 − 2x2)h

=

limh→0−4xh − 2h2

h= −4x .

4.x3

5.√

x

f ′(x) = limh→0 f (x + h) − f (x)h

= limh→0√

x + h − √xh

=

limh→0

√x + h − √x

h·(√

x + h + √x√x + h + √x

)= ( 1

2

)x−1/2.

6.x−1

7. Let f (x) = 1x + 1 .

(a) Show that

f (x + h) − f (x)h

= − 1(x + 1)(x + 1 + h) .

(b) Find f ′(x) by evaluating the limit.(c) Find the equations of the tangent lines to the graph of f at x = .5 and x = 6.

Let f (x) = 1x + 1 .

(a) Thenf (x + h) − f (x)

h=

1

x + h + 1 −1

x + 1h

=−h

(x + h + 1)(x + 1)h

=−1

(x + h + 1)(x + 1) .

(b) Therefore, f ′(x) = limh→0

f (x + h) − f (x)h

= limh→0

−1(x + h + 1)(x + 1) = −

1

(x + 1)2 .(c) For x = 1

2, we have f ′( 1

2) = − 4

9. Thus the tangent line to the graph of f is

y = f ( 12) + f ′( 1

2)(x − 1

2) = 2

3− 4

9(x − 1

2) or y = 8

9− 4

9x .

For x = 6, we have f ′(6) = − 149 . Hence the tangent line to the graph of f isy = f (6) + f ′(6)(x − 6) = 17 − 149 (x − 6) or y = 1349 − x49 .8.

Let f (x) = x−1/2.(a) Show that

f (x + h) − f (x)h

=√

x − √x + hh√

x√

x + h(b) Find f ′(x) by evaluating the limit.

In Exercises 9–17, use the Power Rule to find the derivatives of the following functions.

9. x4

d

dx

(x4) = 4x3.

10.x−4

11. x2/3

d

dx

(x2/3

) = 23

x−1/3.12.

x−2/313. x−10

d

dx

(x−10

) = −10x−11.


14.x−9/5

15. 5

The derivative of a constant is always 0.16.

x√

17

17. x−π2

ddx

(x−π2) = −π 2x−π2−1

18.Find f ′(x) where f (x) = x3 + 3x − 9.

19. Find f ′(x) where f (x) = 6x4 + 7x2/3.Let f (x) = 6x4 + 7x2/3. Then f ′(x) = 24x3 + 14

3x−1/3.

20.Find the equation for the tangent line to the graph of f (x) = 3√x + 8x at x = 9.

In Exercises 21–36, calculate the derivative f ′(x) of each function.

21. x3 + x2d

dx

(x3 + x2) = 3x2 + 2x .

22.2x3 − 3x2 + 2x

23. 2x3 − 10x−1d

dx

(2x3 − 10x−1) = 6x2 + 10x−2.

24.x5 − 7x2 + 10x + 9

25. 4x4 + 38x2 − 14

d

dx

(4x4 + 3

8x2 − 14

)= 16x3 + 3

4x .

26.x1/3

27. 8x1/4

d

dx

(8x1/4

) = 2x−3/4.28. 6

√x + 1√

x29. 7x−3 + x2

d

dx

(7x−3 + x2) = 2x − 21x−4.

30.(x + 1)3 (Hint: expand)

31. (5x + 1)2d

dx

((5x + 1)2) = d

dx

(25x2 + 10x + 1) = 50x + 10.

32.(2x + 3)3

33. (3x − 1)(2x + 1)d

dx((3x − 1)(2x + 1)) = d

dx

(6x2 + x − 1) = 12x + 1.

34.x3(1 − x)

35.√

x(x + 1)d

dx

(x1/2 (x + 1)) = d

dx

(x3/2 + x1/2) =3

2x1/2 + 1

2x−1/2.

36. √x

(1 + 1

x

)37. Find d f /dC where f (C) = 3C2/3.

With f (C) = C2/3, we have d fdC

=23

C−1/3.38.

Find d P/dV where P = 7V

.


39. Find ds/dt where s(t) = 4t − 16t2.With s = 4t − 16t2, we have ds

dt= 4 − 32t .

40.Find d f /dt where f (t) = √t .

41. Find d f /dt where f (t) = 7.With f (t) = 7, we have d f

dx= 0.

42.

Find d f /dx where f (x) =(√

x + 1√x

)2.43. Find d f /dy where f (y) = y5.3.

Let f (y) = y5.3. Then d f/dy = 5.3y4.3.44.Find d f /dx where f (x) = xπ .

In Exercises 45–52, use the limit definition to compute the derivative.

45. f (x) = √x + 2Let f (x) = √x + 2. Then

f ′(x) = limh→0

f (x + h) − f (x)h

= limh→0

√x + h + 2 − √x + 2

h

= limh→0

h

h(√

x + h + 2 + √x + 2) = limh→01√

x + h + 2 + √x + 2= 1

2√

x + 2 .46.

f (x) = x−347. f (x) = 1√

x

Let f (x) = 1√x

. Then

f ′(x) = limh→0

f (x + h) − f (x)h

= limh→0

1√x+h − 1√x

h= lim

h→0

1x+h − 1x

h(

1√x+h + 1√x

)

= limh→0

−h(x+h)x

h(

1√x+h + 1√x

) = limh→0

−1(x + h)x

(1√

x + h +1√x

)

= −1(x2) 2√x

= −12

x−3/2.

48.h(s) = 1

(s + 1)249. f (x) = x−3/2Let f (x) = x−3/2. Then

f ′(x) = limh→0

f (x + h) − f (x)h

= limh→0

(x + h)−3/2 − x−3/2h

= limh→0

(x + h)−3 − x−3h ((x + h)−3/2 + x−3/2) = limh→0

x3−(x+h)3x3(x+h)3

h ((x + h)−3/2 + x−3/2)

= limh→0

−3x2h−3xh2−h3x3(x+h)3

h ((x + h)−3/2 + x−3/2) = limh→0−3x2 − 3xh − h2

x3(x + h)3 ((x + h)−3/2 + x−3/2)= −3x

2

x6 (2x−3/2)= −3

2x−5/2.


50.f (z) = 1

2z + 151. g(t) =√

t2 + 1Let g(t) = √t2 + 1. Then,

g′(t) = limh→0

g(t + h) − g(t)h

= limh→0

√(t + h)2 + 1 − √t2 + 1

h

= limh→0

√(t + h)2 + 1 − √t2 + 1

h·√

(t + h)2 + 1 + √t2 + 1√(t + h)2 + 1 + √t2 + 1

= limh→0

2t + h√(t + h)2 + 1 + √t2 + 1 =

t√t2 + 1 .

52.g(t) = (t + 1)3/2

53. Sketch a graph of the function f (x) = x − 3x2. Calculate f ′(x) and determine the valuesof x for which the tangent line is horizontal.

Let f (x) = x − 3x2. Solve f ′(x) = 1 − 6x = 0 to obtain x = 16. This is the value at which

the tangent line is horizontal.

0

0.04

0.08

0.1 0.2 0.354.

Find the points on the curve y = x2 + 3x − 7 at which the slope of the tangent line is equalto 4.55. Find the points on the curve y = x3 + 3x2 + 1 at which the slope of the tangent line isequal to 15.

Let y = x3 + 3x2 + 1. Solving dy/dx = 3x2 + 6x = 15 or x2 + 2x − 5 = 0 yieldsx = −2 ±

√4 + 20

2= −1 ± √6.

56.Let f (x) = ax2 + bx + c be a quadratic polynomial (assume a �= 0). Show that thereexists a unique value of x at which the tangent line is horizontal. Find this value of x .57. R & W Of the two functions f and g in Figure 1, which is the derivative of the other?Justify your answer.

f (x)

g(x)

1

2

Ð1

Figure 1

g(x) is the derivative of f (x). For f (x) the slope is negative for negative values of x untilx = 0, where there is a horizantal tangent, and then the slope is positive for positive valuesof x . Notice that g(x) is negative for negative values of x , goes through the origin at x = 0,and then is positive for positive values of x .

58.Match the functions (a)–(c) with their derivatives on the right.

Figure 2

59. (a) Graph the functions f (x) = x2 − 5x + 4 and g(x) = −2x + 3.(b) Find the point on the graph of f where the tangent is parallel to the graph of g.


(a)–2

0

2

4

1 2 3

(b) Let f (x) = x2 − 5x + 4 and g(x) = 3 − 2x . Solve f ′(x) = 2x − 5 = −2 = g′(x) toobtain x = 3

2.

60.

GU Let f (x) = 3x and g(x) = 2x .(a) Plot the graphs of f and g on the same set of axes using a GU if necessary.(b) Use the plot to determine the values of x where f ′(x) > g′(x).

61. Determine coefficients a and b such that the polynomial p(x) = x2 + ax + b satisfiesp(0) = 0 and p′(0) = 4.Let p(x) = x2 + ax + b satisfy p(0) = 0 and p′(0) = 4. Since p′(x) = 2x + a, thisimplies 0 = p(0) = b and 4 = p′(0) = a; i.e., a = 4 and b = 0.

62.Find the value of b such that the line 6x + b is tangent to the parabola y = x2.

63. GU(a) Determine coefficients a, b such that the tangent line to the graph of

p(x) = x2 + ax + b at x = 2 is y = 5x − 6.(b) Check your answer by plotting p(x) and y = 5x − 6.

(a) Let p(x) = x2 + ax + b. The tangent line to p at x = 2 isy = p(2) + p′(2)(x − 2) = 4 + 2a + b + (4 + a)(x − 2) = (4 + a)x + b − 4. In orderfor the line y = 5x − 6 to be tangent to p at x = 2, we must have(4 + a)x + b − 4 = 5x − 6. Equate coefficients: 4 + a = 5 and b − 4 = −6. Thisimplies a = 1 and b = −2.

(b)

0

10

1 2 3 4

In Exercises 64–69, find the points c (if any) such that f ′(c) does not exist.64. |x − 1|65. x |x |

Here is the graph of f (x) = x |x |. Its derivative exists everywhere.

–4

–2

2

4

–2 –1 1 2

66. [x]


67. x2/3

Here is the graph of f (x) = x2/3. Its derivative does not exist at x = 0. At that value of x ,the graph is not smooth (notice the sharp corner or “cusp”).

0

1

–2 268. |x − 1|269. |x2 − 1|

Here is the graph of f (x) = ∣∣x2 − 1∣∣. Its derivative does not exist at x = −1 or at x = 1.At these values of x , the graph is not smooth (notice the sharp corners).

0

2

–2 2

Further Insights and Challenges70.

Show that area of the triangle bounded by the tangent line to the graph of y = x−1 at apoint x = a and the coordinate axes does not depend on the value of a.71. The parabola, ellipse, and hyperbola were given their names by the Greek mathematicianApollonius of Prega (b. 262 BCE) in his influential treatise on conics. Prove the followingTheorem of Apollonius: the tangent to the parabola y = x2 at x = a intersects the x-axis atthe midpoint between the origin and (a, 0). Draw a diagram.

Let f (x) = x2. The tangent line to f at x = a isy = f (a) + f ′(a)(x − a) = a2 + 2a(x − a) or y = 2ax − a2. The x-intercept of this line(where y = 0) is a2 , which is halfway between the origin and the point (a, 0).72.R & W Apollonius’ Theorem can be generalized.

(a) Show that the tangent to y = x3 at x = a intersects the x-axis at the point x = 23 a.(b) Formulate the general statement for the graph of y = xn and prove this statement.

73. R & W Let f (x) be a differentiable function and set g(x) = f (x + c), where c is aconstant. Recall that the graph of g(x) is obtained by shifting the graph of f (x) to the left cunits.(a) Use the limit definition to show that g′(x) = f ′(x + c).(b) Explain the result of (a) in words based on the graphical interpretation.

(a) Let g(x) = f (x + c). Using the limit definition,g′(x) = limh→0 g(x + h) − g(x)

h= limh→0 f ((x + h) + c) − f (x + c)

h=

limh→0f ((x + c) + h) − f (x + c)

h= f ′(x + c).

(b) The graph of g(x) is obtained by shifting f (x) to the left by c units. This implies thatg′(x) is equal to f ′(x) shifted to the left by c units which happens to be f ′(x + c).Therefore, g′(x) = f ′(x + c).

74.

Let P = (a, b) be a point on the graph of f (x) = x3 − 3x + 1.(a) Show that if the slope of the tangent line at x = a is negative, then −1 ≤ a ≤ 1.(b) Show that for every number m > −3, there are precisely two points on the graph at

which the slope of the tangent line is equal to m.


75. Negative Exponents Let f (x) = x−n , where n is a positive integer.(a) Show that the difference quotient for f is equal to

−1xn(x + h)n ·

(x + h)n − xnh

.

(b) Prove that f ′(x) = −x−2n(d/dx)xn.(c) Use the Power Rule for xn to show that the Power Rule also holds for x−n . Derive the

Power Rule for negative exponents.

Let f (x) = x−n where n is a positive integer.(a) The difference quotient for f is

f (x + h) − f (x)h

= (x + h)−n − x−nh

=1

(x+h)n − 1xnh

=xn−(x+h)nxn (x+h)n

h

= −1xn(x + h)n

(x + h)n − xnh

.

(b) Therefore,

f ′(x) = limh→0

f (x + h) − f (x)h

= limh→0

−1xn(x + h)n

(x + h)n − xnh

= limh→0

−1xn(x + h)n limh→0

(x + h)n − xnh

= −x−2n ddx

(xn) .

(c) From (b), we continue: f ′(x) = −x−2n ddx

(xn) = −x−2n · nxn−1 = −nx−n−1. Since nis a positive integer, k = −n is a negative integer and we haved

dx

(xk) = d

dx

(x−n

) = −nx−n−1 = kxk−1; i.e. ddx

(xk) = kxk−1 for negative integers

k.76.

Let L be a tangent line to the hyperbola xy = 1 in the first quadrant. Show that the point oftangency is the midpoint of the portion of L lying in the first quadrant. Illustrate with agraph.

77. GU(a) Use a GU to plot the graph of f (x) = (4 − x2/3)3/2 (the “asteroid”).(b) Let L be a tangent line to a point on the graph in the first quadrant. Show that the

portion of L in the first quadrant has a constant length 8.

(a) Here is a graph of the asteroid.

–10

0

10

–10 10

(b) Let f (x) = (4 − x2/3)3/2. Since we have not yet encountered the Chain Rule, we useMaple throughout this exercise. The tangent line to f at x = a is

y = (4 − a2/3)3/2 −√

4 − a2/3(x − a)a1/3

The y-intercept of this line is the point P =(

0, 4√

4 − a2/3)

, its x-intercept is the

point Q = (4a1/3, 0), and the distance between P and Q is 8.78.

Infinitely Rapid Oscillations Let

f (x) ={

x · sin( 1x) if x �= 0

0 if x = 0(a) Show that f is continuous at 0 (use the limit definition of continuity).(b) Show that f ′(0) does not exist.

3.3 Product and Quotient Rules 19

79. A Discontinuous Derivative Let

g(x) ={

x2 · sin( 1x) if x �= 0

0 if x = 0(a) Show, using the limit definition, f ′(0) exists.(b) Calculate f ′(x) for x �= 0 and show that limx→0 f ′(x) does not exist.(c) Conclude that f ′(x) is not continuous at x = 0.

(a) f ′(0) = limh→0

f (0 + h) − f (h)h

= limh→0

f (h) − f (h)h

= limh→0

0

h= 0.

(b) f ′(x) = limh→0

f (x + h) − f (h)h

= limh→0

(x + h)2 sin( 1x+h ) − h2sin(1/h)

h= · · · Already

we see that we run into a problem with the tern sin(1/h) since sin(1/0) is undefined.(c) This follows from the fact that the limit does not exist as x → 0.

80.Let P = (a, f (a)) and Q = (b, f (b)) be points on the graph of y = 1

3x3 − x2. Show that if

the tangent lines at P and Q are parallel, then a + b = 2.

3.3 Product and Quotient Rules


1. Are the following statements true or false? If false, state the correct version.(a) The notation f · g denotes the function whose value at x is f (g(x)).(b) The notation f/g denotes the function whose value at x is f (x)/g(x).(c) The derivative of the product is the product of the derivatives.

2. Are the following equations true or false? If false, state the correct version.

(a)d

dx( f g)

∣∣x=4 = f (4)g′(4) − g(4) f ′(4)

(b)d

dx( f/g)

∣∣x=4 = [ f (4)g′(4) + g(4) f ′(4)]/g(4)2

(c)d

dx( f g)

∣∣x=0 = f (0)g′(0) + g(0) f ′(0)

3. Suppose that f (1) = 0 and f ′(1) = 2. What is the value of g(1) if ( f g)′(1) = 10?4. What is the value of ( f g)′(a), assuming that f (a) and g(a) are 0?

5. What is the derivative of f/g at x = 1 if f (1) = f ′(1) = g(1) = 2, and g′(1) = 4?

Exercises

In Exercises 1–4, use the Product Rule to calculate the derivative.

1. x(x2 + 1)Let f (x) = x(x2 + 1). Then f ′(x) = (x2 + 1) + x(2x) = 3x2 + 1

2. √x(1 − x4)

3. (t2 + 1)(t + 9)Let f (t) = (t2 + 1)(t + 9). Then f ′(t) = 2t (t + 9) + (t2 + 1) = 3t2 + 18t + 1.

4.(x−1/2 + 2x)(7 − x−1)


In Exercises 5–8, use the Quotient Rule to calculate the derivative.

5.x

x − 2Let f (x) = x

x−2 . Then f′(x) = (x−2)−x

(x−2)2 = −2(x−2)2 .6.x + 4

x2 + x + 17. t2 + 1

t2 − 1

Let f (t) = t2 + 1

t2 − 1 . Then f′(t) =

(t2 − 1) (2t) − (t2 + 1) (2t)

(t2 − 1)2 = −4t

(t2 − 1)2 .8.

t2√t + t9. Calculate the derivative of f (t) = (2t + 1)(t2 − 2) in two ways: once using and once not

using the Product Rule.

Using the product rule: Let f (t) = (2t + 1)(t2 − 2). Then,f ′(t) = (2t + 1)(2t) + (t2 − 2)(2) = 6t2 + 2t − 4. Multiplying out first: Letf (t) = 2t3 + t2 − 4t − 2. Then, f ′(t) = 6t2 + 2t − 4.

10.Let f (x) = g(x) = x and show that ( f/g)′ does not equal f ′/g′.

In Exercises 11–38, calculate the derivative using the appropriate rule or combination of rules.

11. (x4 − 4)(x2 + x + 1)Let f (x) = (x4 − 4) (x2 + x + 1). Thenf ′(x) = (x4 − 4) (2x + 1) + (x2 + x + 1) (4x3) = 6x5 + 5x4 + 4x3 − 8x − 4.

12.3√

x − 2x13. (x

2 + x + 2)(x3 − 7x − 1)Let f (x) = (x2 + x + 2) (x3 − 7x − 1). Thenf ′(x) = (x2 + x + 2) (3x2 − 7)+ (x3 − 7x − 1) (2x + 1)= 5x4 + 4x3 − 15x2 − 16x − 15.

14.x4 − 4x2 − 515. x + 1x1/2 + 1

Let f (x) = x + 1√x + 1 . Then f

′(x) =(√

x + 1) (1) − (x + 1)( 12√x )(√x + 1)2 =

x + 2√x − 12√

x(√

x + 1)2 .16.

x4 + 2x + 1x + 117.

√x + 1√x − 1

Let f (x) =√

x + 1√x − 1 . Then f

′(x) =(√

x − 1) 12√x − (√x + 1) 12√x(√x − 1)2 = −

1√x(√

x − 1)2 .18.

x3

x + 119. (x7 + 3x2 − 2)(x5 − 4x3)

Let f (x) = (x7 + 3x2 − 2) (x5 − 4x3). Thenf ′(x) = (x7 + 3x2 − 2) (5x4 − 12x2)+ (x5 − 4x3) (7x6 + 6x)= 12x11 − 40x9 + 21x6 − 70x4 + 24x2.

20.z2 + z − 1z2 + z + 1


21.5z − 94 − 4zLet f (z) = 5z − 9

4 − 4z . Then

f ′(z) = (4 − 4z) (5) − (5z − 9) (−4)(4 − 4z)2 = −

1

(z − 1)2 .

22. x(x4 + x2)(x7 + 1)23. (x2 + 9)(x + x−1)

Let f (x) = (x2 + 9) (x + 1x

). Then

f ′(x) = (x2 + 9) (1 − 1x2

)+(

x + 1x

)(2x) = 3x2 + 10 − 9

x2

or3x4 + 10x2 − 9

x2.

24.(√

x − 1)(x + 1)25. 31/2 · 51/2

Let f (x) = √3√5. Then f ′(x) = 0, since f (x) is a constant function!26.

x2(√

x − 1)(x + 1)27.

x

x + x−1

Let f (x) = xx + x−1 =

x2

x2 + 1 , for x �= 0. Then

f ′(x) =(x2 + 1) (2x) − x2 (2x)

(x2 + 1)2 =2x

(x2 + 1)2 , for x �= 0.28.(x + 1) x

x + x−129. π 2(x − 1)Let f (x) = π 2(x − 1). Then f ′(x) = π 2.

30. (x + 1x − 1

)(x

x2 + 1)

31. (x + 3)(x − 1)(x − 5)Let f (x) = (x + 3)(x − 1)(x − 5). Thenf ′(x) = (x + 3) ((x − 1)(1) + (x − 5)(1)) + (x − 1)(x − 5)(1) = 3x2 − 6x − 13.Alternatively, f (x) = (x + 3) (x2 − 6x + 5) = x3 − 3x2 − 13x + 15, whencef ′(x) = 3x2 − 6x − 13.

32.x(x2 + 1)(x + 4)

33.√

x(x2 + 1)(x1/3 − 3)Let f (x) = √x (x2 + 1) (x1/3 − 3) = x17/6 − 3x5/2 + x5/6 − 3x1/2. Thenf ′(x) = 17

6x11/6 − 15

2x3/2 + 5

6x−1/6 + 3

2x−1/2.

34.x−1 + x−2√

x + 135. z2 + 2z − 1

z2 − 1z + 1 Hint: simplify first

Let f (z) = z2 + 2z − 1

z2 − 1z + 1 = z

2 + 2, for z �= ±1. Then f ′(z) = 2z, for z �= ±1.36.

d

dx

(zx2(xz2 + 1)) (z constant)

37.d

dx

((ax + b)(abx2 + 1)) (a, b constants)

Let f (x) = (ax + b) (abx2 + 1). Thenf ′(x) = (ax + b) (2abx) + (abx2 + 1) (a) = 3a2bx2 + a + 2ab2x .


38.d

d RR(R + r R + r2) (r constant)

39. The curve y = 1x2 + 1 is called the witch of Agnesi, after the Italian mathematician Maria

Agnesi (1718–1799). This strange name is the result of a mistranslation of the originalItalian name la versiera, which means “rope that turns a sail.”(http://www-groups.dcs.st-andrews.ac.uk/ history/Mathematicians/Agnesi.html) Find theequations of the tangent lines to the curve at x = ±1.

1 2 3

1

–1–2–3

Figure 1 Graph of the witch of Agnesi.

Let f (x) = 1x2 + 1 . Then f

′(x) = − 2x(x2 + 1)2 .

At x = −1, the tangent line is y = f (−1) + f ′(−1)(x + 1) = 12

+ 12(x + 1) = 1

2x + 1

or y = 12 x + 1.At x = 1, the tangent line is y = f (1) + f ′(1)(x − 1) = 12 − 12 (x − 1) = 1 − 12 x ory = 1 − 1

2x .

In Exercises 40–43, use the function values

f (4) f ′(4) g(4) g′(4)2 −3 5 −1

40.Find the derivative of f g and f/g at x = 4.

41. Calculate F ′(4), where F(x) = x f (x).Let F(x) = x f (x). Then F ′(x) = x f ′(x) + f (x), whenceF ′(4) = 4 f ′(4) + f (4) = 4(−3) + 2 = −10.42.Calculate G ′(4), where G(x) = xg(x) f (x).

43. Calculate H ′(4), where H(x) = xg(x) f (x)

.

Let H(x) = xg(x) f (x)

. Then H ′(x) = g(x) f (x) · 1 − x (g(x) f′(x) + f (x)g′(x))

(g(x) f (x))2.

Therefore, H ′(4) = (5)(2) − 4 · ((5)(−3) + (2)(−1))((5)(2))2

= 78100

= 3950

.44.

Use the Quotient Rule to calculate F ′(0) where

F(x) = x29 + 14x12 − 16x11 + 17x8 + 3x − 2

x11 + 7x4 − 2x − 3Hint: do not compute F ′(x). Instead, write F as f/g and express F ′(0) in terms of thef (0), f ′(0), g(0), g′(0).

45. Calculate F ′(0) where

F(x) = (1 + x + x1/2 + x3/2) 3x5 + 5x4 + 5x + 1

8x9 − 7x4Hint: see hint for previous exercise.

The function given is not defined at x = 0. Accordingly, its derivative cannot exist there.46.Show that if f is a differentiable function, then

d

dx(x f (x)) = x f ′(x) + f (x)

47. Use the Product rule to show that ( f 2)′ = 2 f · f ′.Let g = f 2 = f f . Then g′ = ( f 2)′ = ( f f )′ = f f ′ + f f ′ = 2 f f ′.48.Show that ( f 3)′ = 3 f 2 · f ′. Hint: write f 3 as f · f 2.


Further Insights and Challenges

49. Let f, g, h be differentiable functions. Show that ( f gh)′(x) is equal to

f (x)g(x)h′(x) + f (x)g′(x)h(x) + f ′(x)g(x)h(x)Hint: write f gh as ( f g)h and apply the Product Rule twice.

Let p = f gh. Then p′ = ( f gh)′ = f (gh′ + hg′) + gh f ′ = f ′gh + f g′h + f gh′.50.

Prove the Quotient Rule using the limit definition of the derivative.51. Derivative of the Reciprocal Let f (x) be a function.

(a) Show that the difference quotient for 1/ f (x) is equal to

f (x) − f (x + h)h f (x) f (x + h)

(b) Show that 1/ f (x) is differentiable for all x such that f (x) is differentiable andf (x) �= 0, and

d

dx

(1

f (x)

)= − f

′(x)f 2(x)

(c) Use (b) and the Product Rule to derive the Quotient Rule. Hint: Write f/g as f · (1/g).(d) Show that (b) can also be proved using the Quotient Rule.

Let f (x) be differentiable.(a) Let g(x) = 1/ f (x). The difference quotient for g is

g(x + h) − g(x)h

=1

f (x + h) −1

f (x)h

=f (x) − f (x + h)

f (x + h) f (x)h

= f (x) − f (x + h)h f (x) f (x + h) =

−1f (x) f (x + h)

f (x + h) − f (x)h

(b) Provided f (x) �= 0, then

limh→0

g(x + h) − g(x)h

= limh→0

( −1f (x) f (x + h)

f (x + h) − f (x)h

)= − f

′(x)

( f (x))2.

Because the limit exists, g(x) = 1/ f (x) is differentiable for all x .(c) Let f/g = f · (1/g). From Exercise 51, we have(

f

g

)′=(

f ·(

1

g

))′= f · −g

′

g2+ 1

g· f ′ = g f

′ − f g′g2

.

(d) Let g(x) = 1/ f (x), using the quotient rule, g′(x) = f′(x) − f (x) · 0

f 2(x)= f

′(x)f 2(x)

52.

In 1748, Maria Agnesi published a book on calculus that contained a proof of the QuotientRule using the Product Rule. Let f and g be differentiable functions and assume that thequotient h = f/g is differentiable.

(a) Compute the derivative of hg = f using the Product Rule.(b) Solve for h′.

53. R & W We say that a is a multiple root of a polynomial f (x) if (x − a)2 is a factor off (x). In other words, f (x) = (x − a)2g(x), where g(x) is also a polynomial. Show that ifa is a multiple root of f , then f ′(a) = 0.Assume that f (x) = (x − a)2g(x). Thenf ′(x) = (x − a)2g′(x) + g(x)2(x − a) = (x − a)2g′(x) + g(x)(2x − 2a). Therefore,f ′(a) = (a − a)2g′(a) + g(a)(2a − 2a) = 0.

54.

Let f (x) be a polynomial.(a) Show that a is a multiple root of f (x) if a is a root of both f (x) and f ′(x). Hint: use

the fact from algebra that a is a root of a polynomial g(x) if and only ifg(x) = (x − a)h(x) for some polynomial h(x).

(b) Show that −1 is a multiple root of x5 + 2x4 − 4x3 − 8x2 − x + 2.(c) Determine if 1 is a multiple root of the polynomials:

i. x4 − 2x3 + x2 + x − 1ii. 2x6 − 4x5 + x4 + 2x3 − 2x + 1


The next problem uses the Product Rule to prove the Power Rule for positive integer exponents.It assumes that you are familiar with proof by induction.

55. The Power Rule Revisited Let n be a positive integer.(a) Show directly that the Power Rule holds for n = 1.(b) Write xn as x · xn−1 and use the Product Rule to show

d

dxxn = x d

dxxn−1 + xn−1

(c) Use induction to prove that the Power Rule holds for all n.

Let k be a positive integer.

(a) If k = 1, then xk = x . Note that ddx

(x1) = d

dx(x) = 1 = 1x0. Hence the Power Rule

holds for k = 1.(b) For a positive integer n ≥ 2, observe that

d

dx(xn) = d

dx

(x · xn−1) = x d

dx

(xn−1

)+ xn−1 ddx

(x) = x ddx

xn−1 + xn−1.(c) By (a) the Power Rule holds for k = 1. Assume it holds for k = n where n ≥ 2. Then

for k = n + 1, we haved

dx

(xk) = d

dx

(xn+1

) = ddx

(x · xn) = x ddx

(xn) + xn ddx

(x)

= x · nxn−1 + xn · 1 = (n + 1)xn = kxk−1Accordingly, the Power Rule holds for all positive integers by induction.

3.4 Rates of Change


1. Pressure in a tank of water (measured in atmospheres) is a function of depth. What unitscould be used to measure the rate of change of pressure with respect to depth?

2. The concentration of nitrogen dioxide is measured in moles per liter. What units are used tomeasure the reaction rate, which is the rate of change of concentration?

3. How do we justify the definition of instantaneous rate of change as the derivative?

4. In his Lectures on Physics, Nobel laureate Richard Feynmann (1918–1988) uses the follow-ing dialogue to make a point about instantaneous velocity:

Policeman: “My friend, you were going 75 miles an hour.”

Driver: “That’s impossible, sir, I was traveling for only seven minutes.”

What is wrong with the driver’s response?

5. Two trains travel from New York to Boston in 3 12

hours. The first train travels at a constantvelocity of 60 mph, but the velocity of the second train varies. What was the second train’saverage velocity during the trip?

6. True or False: the graph of velocity for a falling object is a parabola.

7. Could marginal cost be negative in a real situation?

3.4 Rates of Change 25

Exercises

1. The distance (in feet) traveled by a particle at time t (in minutes) is s(t) = 110

t4 − 16t3 + 1.

Calculate the particle’s velocity at t = 5.First compute s ′(t) = 2

5t3 − 1

2t2, then evaluate s ′(5) = 75/2 = 37.5 ft/min.

2.

Figure 1 shows the graph of distance versus time for a train trip.(a) Estimate the average velocity over the interval [.5, 1].(b) Is the average velocity greater during [1, 2] or [2, 3]?(c) At what time does the train attain its maximum (instantaneous) velocity?

Figure 1 Graph of distance versus time for a train trip.

3. A stone is tossed vertically upward with an initial velocity of 25 ft/sec from the top of a30-ft building.(a) What is the height of the stone after .25 seconds?(b) Find the velocity of the stone after 1 second.(c) When does the stone hit the ground?

We employ Galileo’s formula, s(t) = s0 + v0t − 12 gt2 = 30 + 25t − 16t2, where the time tis in seconds (s) and the height s is in feet (ft).(a) The height of the stone after .25 seconds is s(.25) = 35.25 ft.(b) The velocity at time t is s ′(t) = 25 − 32t . When t = 1, this is −7 ft/s.(c) When the stone hits the ground, its height is zero. Solve 30 + 25t − 16t2 = 0 to obtain

t = 25 ±√

2545

32or t ≈ 2.36 s.

(The other solution, t ≈ −0.79, we discard since it represents a time before the stonewas thrown.)

4.The height (in feet) of a skydiver at time t (in seconds) after opening his parachute ish(t) = 2000 − 15t ft. Find the skydiver’s velocity.5. The velocity (in cm/s) of blood molecules flowing through a capillary of radius .008 cmdepends on their distance r from the center of the capillary, according to the formulav = .001(64 · 10−6 − r2). Find the rate of change of velocity as a function of distance whenr = .004 cm.The velocity of the blood molecules is v′(r) = −.002r . When r = .004 cm, this rate is8 · 10−6cm2/s.

6.The earth exerts a gravitational force of strength F(r) = 2.99 · 1016/r2 (in Newtons) on anobject of mass 75 kg, where r is the distance (in meters) from the center of the earth. Findthe rate of change of force with respect to distance at the surface of the earth (assume theradius of the earth is 6.77 · 106 m).

7. The escape velocity at a distance r from the center of the earth is vesc = 2.82 · 107 · r−1/2m/s. Calculate the rate at which escape velocity changes with respect to distance at thesurface of the earth (assume the radius of the earth is 6.77 · 106 m).The rate that escape velocity changes is v′esc(r) = − 12 r−3/22.82 · 107.Therefore,v′esc(6.77 · 106) = − 12 (6.77 · 106)−3/22.82 · 107 = −8 · 1014.8.A 1.5-volt battery with internal resistance r = .5 ohms is connected to a small light whoseresistance R can be controlled by a dial. The power delivered to the light is

P = 2.25R(R + .5)2 watts

Find the rate of change of power with respect to resistance when R = 3 and R = 5 ohms.

9. Suppose that the height of a jumping frog is h(t) = 2.2t − 2.5t2.(a) What is the frog’s velocity at t = .3?(b) When is the frog’s velocity equal to 0?(c) What is the maximum height of the frog?

(a) The velocity is h′(t) = 2.2 − 5t . Therefore, h′(.3) = .7.(b) Solving for h′(t) = 2.2 − 5t = 0, we obtain t = .44.(c) The maximum height the frog reaches is when the velocity is zero, which happens to

be at time t = .44. Therefore, the maximum height is h(.44) = 9.196.10.

Suppose that the temperature (in degrees Fahrenheit) of a hot object as it cools in a freezeras a function of time (in minutes) is T (t) = 3

4t2 − 30t + 340 for 0 ≤ t ≤ 20. At what rate is

the object cooling after 10 minutes (give correct units)?11. By Faraday’s law, if a conducting wire of length meters moves at velocity v m/s in a

magnetic field of strength B (in teslas), a voltage of size V = −Bv is induced in the wire.Assume that B = 2 and = .5.


(a) Find the rate of change of V with respect to v.(b) Find the rate of change of V with respect to time t if v = 4t + 9.

(a) Assuming that B = 2 and l = 5, V = −2(.5)v. Therefore, dVdv

= −2(.5) = −1.(b) If v = 4t + 9, then V = −2(.5)(4t + 9). Therefore, dV

dt= −2(.5)(4) = −4.

12.

Suppose the height (in feet) of a helicopter at time t (in minutes) is s(t) = −3t3 + 400t for0 ≤ t ≤ 15.

(a) Plot the graphs of height s(t) and the velocity v(t).(b) Find the velocity of the helicopter at times t = 6 and t = 7.(c) When is the velocity of the helicopter equal to zero and what is its height at that time?(d) What is the maximum height of the helicopter?

13. R & W Table 1 gives total U.S. population during each month of 1999 as determined bythe U.S. Department of Commerce.(a) Estimate P ′(t) for each of the months January-November.(b) Plot these data points for P ′(t) and connect the data points by a smooth curve.(c) Write a newspaper headline describing the information contained in this plot.

Table 1

t P(t) in thousands t P(t) in thousands

January 271841 July 272945February 271987 August 273197March 272142 September 273439April 272317 October 273672May 272508 November 273891June 272718 December 274076

The table in the text gives the growing population P(t) of the United States.

(a) Here are estimates of P ′(t) in thousands/month for January–November.

t Jan Feb Mar Apr May Jun Jul Aug Sep Oct NovP ′(t) 146 155 175 191 210 227 252 242 233 219 185

(b) Here is a plot of these estimates (1 = Jan, 2 = Feb, etc.)P’(t)

0

100

200

300

5 10

(c) “U.S. Growth Rate Declines After Midsummer Peak”14.

Let A denote the area of a square whose side has length s.(a) What is the rate of change of A with respect to s?(b) What is the rate of change of A with respect to the perimeter?

15. Suppose that a cylinder has height equal to the radius r of its base. Calculate the rate ofchange of the volume of the cylinder with respect to r .

The volume of the cylinder is V = πr2h = πr3. Thus dV/dr = 3πr2.16.

Find the rate of change of the volume of a cube with respect to the length s of a side.17. Find the rate of change of the volume of a cube with respect to its surface area.

The volume of a cube of side length s is V = s3. Its surface area is A = 6s2. Thuss = 1√

6A1/2 and V =

(1√6

A1/2)3 = 1

6√

6A3/2. Hence dV/d A = 1

4√

6A1/2 = √6A/24.

18.

The volume of a sphere of radius r is V = 43 πr3 and its surface area is A = 4πr2.(a) Calculate the rate of change of V with respect to radius.(b) Calculate the rate of change of A with respect to diameter. Hint: express the area as a

function of diameter.


19. The position of a moving particle during a 5-second trip is s(t) = t2 − t + 10 cm.(a) What is the average velocity for the entire trip?(b) Is there a time at which the instantaneous velocity of the particle is equal to this

average velocity? If so, find it.

Let s(t) = t2 − t + 10, 0 ≤ t ≤ 5, with s in centimeters (cm) and t in seconds (s).(a) The average velocity over the t-interval [0, 5] is s(5) − s(0)

5 − 0 =30 − 10

5= 4 cm/s.

(b) The (instantaneous) velocity is v(t) = s ′(t) = 2t − 1. Solving 2t − 1 = 4 yields t = 52

s, the time at which the instantaneous velocity equals the stated average velocity.20.

A slingshot is used to shoot a small ball in the air vertically from ground level with aninitial velocity 200 m/sec.

(a) What is the maximum velocity attained by the ball?(b) What is the maximum height of the ball?

21. What is the velocity of an object dropped from a height of 300 m when it hits the ground?

We employ Galileo’s formula, s(t) = s0 + v0t − 12 gt2 = 300 − 4.9t2, where the time t is inseconds (s) and the height s is in meters (m). When the ball hits the ground its height is 0.Solve s(t) = 300 − 4.9t2 = 0 to obtain t ≈ 7.8246. (We discard the negative time, whichtook place before the ball was dropped.) The velocity at impact is v(7.8246) ≈ −76.68m/s. This signifies that the ball is falling at 76.68 m/s.22.It takes a stone 3 seconds to hit the ground when dropped from the top of a building. Howhigh is the building and what is the stone’s velocity upon impact?23. A ball tossed up vertically from ground level returns to earth four seconds later. What wasthe initial velocity of the stone?

Galileo’s formula gives s(t) = s0 + v0t − 12 gt2 = v0t − 4.9t2, where the time t is inseconds (s) and the height s is in meters (m). When the ball hits the ground after 4 secondsits height is 0. Solve 0 = s(4) = 4v0 − 4.9(4)2 to obtain v0 = 19.6 m/s.24.An object is tossed up vertically from ground level and hits the ground T seconds later.Show that its maximum height was reached after T/2 seconds.25. Which of the following statements is true for an object falling under the influence ofgravity near the surface of the earth?(a) The object covers equal distance in equal time intervals.(b) The velocity of a falling object increases by equal amounts in equal time intervals.(c) The derivative of velocity increases with time.

For an object falling under the influence of gravity, Galileo’s formula givess(t) = s0 + v0t − 12 gt2.(a) Since the height of the object varies quadratically with respect to time, it is not true that

the object covers equal distance in equal time intervals.(b) The velocity is v(t) = s ′(t) = v0 − gt . The velocity varies linearly with respect to

time. Accordingly, the velocity decreases (becomes more negative) by equal amountsin equal time intervals. Moreover, its speed (the magnitude of velocity) increases byequal amounts in equal time intervals.

(c) Acceleration, the derivative of velocity with respect to time, is given bya(t) = v′(t) = −g. This is a constant; it does not change with time. Hence it is nottrue that acceleration (the derivative of velocity) increases with time.26.

As an epidemic spreads through a population, the percentage p of infected individuals isknown to satisfy the equation (where time t is measured in days)

dp

dt= 4p − .06 p2

(a) How fast is the epidemic spreading when p = 10% and p = 70%?(b) For which p is the epidemic neither increasing nor decreasing (the rate is 0)?(c) Draw a graph of the derivative dp/dt as a function of p.(d) What is the maximum possible rate of increase of the epidemic and when (for which p)

does this occur?

27. A weight oscillates up and down at the end of a spring. Figure 2 shows the height y of theweight through one cycle of the oscillation. Make a rough sketch of the graph of thevelocity as a function of time.

time

y

Figure 2


Here is the graph of the velocity of a function of time:

time

y

28.Use Figure 3 to sketch the graph of dT /dh, where T is the temperature of the earth’satmosphere and h is altitude.

Figure 3 Graph of atmospheric temperature versus altitude.

29. The population P(t) of a city (in millions) is given by the function

P(t) = (.00005)t2 + (.01)t + 1,where t denotes the number of years since 1990.(a) How large is the population in 1996 and how fast is it growing?(b) When is the population growing at a rate of 12,000 people per year?

Let P(t) = (.00005)t2 + (.01)t + 1 be the population of a city in millions. Here t is thenumber of years past 1990.(a) In 1996 (t = 6 years after 1990), the population is P(6) = 1.0618 million. The rate of

growth of population is P ′(t) = .0001t + .01. In 1996, this corresponds to a growthrate of P ′(6) = .0106 million per year or 10,600 people per year.

(b) When the growth rate is 12,000 people per year (or .012 million per year), we haveP ′(t) = .0001t + .01 = .012, whence t = 20. This corresponds to the year 2010; i.e.,20 years past 1990.

30.

According to Ohm’s law, the voltage V , current I , and the resistance R are related by theequation V = I R, where the units are volts, amperes, and ohms. Assume that V = 12 volts.

(a) Calculate the average rate of change of current with respect to resistance for theinterval from R = 8 to R = 8.1.

(b) Calculate the rate of change of current with respect to resistance when R = 8(specifying the units).

(c) Calculate the rate of change of resistance with respect to current when I = 1.5(specifying the units).

31. The dollar cost of producing x bagels (in units of 1000) is C(x) = 300 + .25x − 5 · 10−10x3.(a) Find the cost of producing x bagels for x = 2000 and x = 4000.(b) Estimate the cost of the 2001st bagel (in other words, estimate C(2001) − C(2000)).

Do the same for x = 4000.

(a) C(2000) = 300 + .25(2000) − 5 · 10−10(2000)3 = 796 dollars.C(4000) = 300 + .25(4000) − 5 · 10−10(4000)3 = 1268 dollars.

(b) C ′(2000) ≈ C(2001) − C(2000). Therefore, C ′(2000) ≈ .244 dollars andC ′(4000) ≈ C(4001) − C(4000) ≈ .226 dollars.

32.

The reproduction rate of the fruit fly Drosophila melanogaster, grown in bottles in alaboratory, decreases as the bottle becomes more crowded. It is found empirically thatwhen a bottle contains p flies, the number of offspring produced by each female each day is

f (p) = (34 − 0.612p)p−0.658

(a) Calculate f (15) and f ′(15).(b) Estimate the decrease in number of offspring produced by each female when p is

increased from 15 to 16.(c) Which is larger, the actual number f (16) − f (15) or the estimate obtained in (b)?

33. Assume that the stopping distance (in feet) is given by the formula F(s) = 1.1s + .03s2,where s is the vehicle’s speed in mph (see Example ??).(a) Calculate F(65) and F ′(65).(b) Estimate the increase in stopping distance if speed is increased from 65 mph to 66 mph.

Let F(s) = 1.1s + .03s2 be as in Example 4.


(a) Then F(65) = 198.25 ft and F ′(65) = 5.00 ft/mph.(b) F ′(65) ≈ F(66) − F(65) is approximately equal to the change in stopping time per 1

mph increase in speed when traveling at 65 mph.34.

Suppose the dollar-cost of producing x video cameras is

C(x) = 500 x − .003 x2 + 10−8x3

(a) Estimate the marginal cost at production level x = 10000.(b) Calculate the actual cost C(10001) − C(10000) of producing an additional camera and

compare with (a).Further Insights and Challenges

35. Let P(t) be the size of a certain animal population in a forest at time t (in months). It isfound experimentally that the rate of change satisfies

d P

dt= 0.2(300 − P)

(a) Is the population growing or shrinking if P = 250? If P = 350?(b) Graph d P/dt as a function of P for 0 ≤ P ≤ 300.(c) Which of the graphs in Figure 4 has the correct shape to be the graph of P(t)?

4 8

100

200

300

100

300

4 8

(a) (b)

200

Figure 4

Let P ′(t) = d P/dt = 0.2(300 − P).(a) Since P ′(250) = 10, the population is growing when P = 250.

Since P ′(350) = 10, the population is shrinking when P = 350.(b) Here is a graph of d P/dt for 0 ≤ t ≤ 300.

0

20

40

60

100 200 300

(c) If P(0) = 285, as in graph (a), then P ′(0) = 3 > 0 and the population is growing,contracting what is depicted in graph ?. Accordingly, graph (a) cannot be the correctshape for P(t).If P(0) = 250, as in graph (b), then P ′(0) = 10 > 0 and the population is growingas depicted. Thus graph (b) has the correct shape for P(t).


In Exercises 36–37, define the average cost per unit at production level x as

Cav(x) = C(x)x

where C(x) is the cost function. Average cost is a measure of the efficiency of the productionprocess.36.

Suppose the cost in dollars of producing alarm clocks is

C(x) = 50x3 − 750x2 + 3740x + 3750where x is in units of one thousand.

(a) Interpret Cav(x) graphically by showing that it is equal to the slope of the line throughthe origin and the point (x, C(x)) on the graph of C(x).

(b) Use the interpretation in (a) to determine which is greater, the marginal cost or averagecost, at x = 2 and at x = 6. Do this graphically, without making any calculations.

(c) Use the formula for C(x) to calculate the marginal cost and the average cost at x =2, 7, 8.

(d) Use the graph to find the production level x0 at which average cost is lowest. What isthe relation between average cost and marginal cost at x0?

Figure 5 Cost function C(x) = 50x3 − 750x2 + 3740x + 3750.

37. Let f (x) = cos x and A(x) = f (x)/x .(a) Sketch the graph of cos x for 0 ≤ x ≤ 2π .(b) Indicate the point where A(x) takes on its smallest value. Hint: Place a ruler on the

graph so that it goes through the point (0, 1) and points (x, cos x). Rotate it until it hasminimum slope.

(c) What happens if you place the ruler on the graph in (a) and let x → 0?

Let C(x) = cos x and A(x) = cos xx

.

(a) Here is a graph of C(x) and A(x) on the same set of axes.

0 1 2 3 4 5 6−2

−1

0

1

2

3

4

C(x) = cos(x)

A(x) = cos(x) / x

(b) We place a ruler on the graph so that it goes through the origin and the point (x, C(x))on the graph (as was done in Exercise ??) and rotate it until it has minimum slope. Thisoccurs around x0 = 2.8.

(c) As we rotate the ruler and let x → 0, the slope in question becomes positively infinite(and hence A(x) → ∞).

38.The tangent lines to the graph of f (x) = x2 get steeper as x increases. At what rate are theslopes of the tangent lines increasing?

3.5 Higher Derivatives


1. An economist who announces that “America’s economic growth is slowing” is making astatement about the GNP (gross national product) as a function of time. Is the second deriva-tive of the GNP positive? What about the first derivative?

3.5 Higher Derivatives 31

2. On December 29, 1993, The New York Times published an article with the headline “Risein California Population Slows to Trail Nation’s Rate.” This statement is poorly wordedbecause it seems to compare a rise, which is the actual change in population, to rate ofchange. Rewrite the headline as a statement about the appropriate derivative of California’spopulation.

3. Rephrase the following as a statement about the appropriate higher derivative: the velocityis still increasing, but not as rapidly as before. Sketch a possible graph of velocity.

4. True or False: the third derivative of position is zero for an object falling to earth under theinfluence of gravity.

5. Which polynomials have the property that f ′′(x) = 0 for all x?

Exercises

In Exercises 1–6, calculate the second and third derivatives of the function.

1. 14

Let f (x) = 14. Then f ′(x) = 0, f ′′(x) = 0, and f ′′′(x) = 0.2.

x4 − 25x2 + 2x3. z − 1

z

Let f (z) = z − z−1. Then f ′(z) = 1 + z−2, f ′′(z) = −2z−3, and f ′′′(z) = 6z−1.4.

x−9/5

5. t2(t2 + t)Let g(t) = t2(t2 + 1) = t4 + t2. Then g′(t) = 4t3 + 2t , g′′(t) = 12t2 + 2, g′′′(t) = 24t .

6.20t4/5 − 6t2/3

7. Find the acceleration at time t = 5 minutes of a helicopter whose height (in feet) iss(t) = −3t3 + 400t .Let s(t) = 400t − 3t3, with t in minutes and s in feet. The velocity isv(t) = s ′(t) = 400 − 9t2 and acceleration is a(t) = −18t . Thus a(5) = −90 ft/min.

8.Calculate the first four derivatives of f (x) = x4/5.

In Exercises 9–24, calculate the derivative indicated.

9. f (4)(1); f (x) = x4Let f (x) = x4. Then f ′(x) = 4x3, f ′′(x) = 12x2, f ′′′(x) = 24x , and f (4) = 24. Thus

f (4)(1) = 24.10.

g′′′(1); g(t) = 4t−311. y ′′′(1); y = 4t−3 + 3t2 − 9t

Let y = 4t−3 + 3t2 − 9t . Then y ′ = −12t−4 + 6t − 9, y ′′ = 48t−5 + 6, and y ′′′ = −240t−6.Hence y ′′′(1) = −240.

12.f (4)(1); f (t) = 6t9 − 2t5

13. h′′′(9); h(x) = √xLet h(x) = √x = x1/2. Then h′(x) = 1

2x−1/2, h′′(x) = − 1

4x−3/2, and h′′′(x) = 3

8x−5/2.

Thus h′′′(9) = 1648

.14.

g′′′(9); g(x) = x−1/2


15. h′′′(16); h(x) = x−3/4Let g(x) = x−3/4. Then g′(x) = (− 43 ) x−7/2, g′′(x) = ( 2116 ) x−11/4, andg′′′(x) = (− 231

64

)x−15/4. Thus g′′′(16) = (−231/64)16−15/4.

16.y ′′(0); y = 1

t2 + t + 117. f ′′(1); f (t) = 1t3 + 1

Let y = 1t3 + 1 . Then

f ′(t) = −3t2

(t3 + 1)2 and f′′(t) =

(t3 + 1)2 (−6t) − (−3t2) · 2 (t3 + 1) (3t2)

(t3 + 1)4 .

Thus f ′′(1) = −24 + 3616

= 34

. (NOTE: To compute the derivative of(t3 + 1)2 = (t3 + 1) (t3 + 1), use the Product Rule.)

18.g′′(1); g(x) = x

x + 119. h′′(1); h(x) = 1√

x + 1

Let h(x) = 1√x + 1 . Then h

′(x) = −12x−1/2(√

x + 1)2 and

h′′(x) =(√

x + 1)2 ( 14x−3/2

)− (− 12x−1/2

) · 2 (√x + 1) ( 12x−1/2

)(√

x + 1)4 .Accordingly, h′′(x)(1) = 1+116 = 18 . (NOTE: To compute the derivative of(√

x + 1)2 = (√x + 1) (√x + 1), use the Product Rule.)20.

y ′′(1); y = x − 1x21. y ′′′(1); y = x − 1x

Let y = x − x−1. Then y ′ = 1 + x−2, y ′′ = −2x−3, and y ′′′ = 6x−4. Hence y ′′′(1) = 6.22.

F ′′(2); F(x) = (x2 + 1)−123. y(4)(1); y = x4 + ax3 + bx2 + cx + d

a, b, c, d constants

Let y = x4 + ax3 + bx2 + cx + d , where a, b, c, d are constants. Theny ′ = 4x3 + 3ax2 + 2bx + c, y ′′ = 12x2 + 6ax + 2b, y ′′′ = 24x + 6a, and y(4) = 24. Hencey(4)(1) = 24.

24.y(3)(0); y = x4 + ax3 + bx2 + cx + da, b, c, d constants25. Find a polynomial f (x) satisfying the equation x f ′′(x) + f (x) = x2. Hint: try a quadraticpolynomial.

Let f (x) = ax2 + bx + c. Then f ′(x) = 2ax + b and f ′′(x) = 2a. Substituting into theequation x f ′′(x) + f (x) = x2 yields ax2 + (2a + b)x + c = x2, an identity in x . Equatingcoefficients, we have a = 1, 2a + b = 0, c = 0, whence a = 1, b = −2, and c = 0.

26.

Which of the following functions satisfy f (k)(x) = 0 for all k ≥ 6?(a) 7x4 + 4 + x−1(b) x3 − 2(c)

√x

(d) 1 − x6(e) x9/5

(f) 2x2 + 3x5

27. Find the equation of the tangent to the graph of f ′ at x = 3, where f (x) = x4.Let f (x) = x4 and g(x) = f ′(x) = 4x3. Then g′(x) = 12x2. The tangent line to g at x = 3is given by y = g(3) + g′(3)(x − 3) = 108 + 108(x − 3) = 108x − 216 ory = 108x − 216.

28.Find a general formula for the nth derivative of f (x) = (x + 1)−1.

3.5 Higher Derivatives 33

29. Find a general formula for the nth derivative of f (x) = x−2.f ′(x) = −2x−3, f ′′(x) = 6x−4, f ′′′(x) = 24x−5, f (4)(x) = 5 · 24(x)−6, . . . . From this wecan conclude that the nth derivative can ve written as f (n)(x) = (−1)n(n + 1)!x−(n+2).

30.Find a general formula for the nth derivative of f (x) = x1/2.

31. Figure 1 shows f, f ′ and f ′′. Determine which is which.

1 2 31 2 31 2 3

(a) (b) (c)

Figure 1

(a) f ′′ (b) f ′ (c) f .32.

The second derivative f ′′ is shown in Figure 2. Determine which of (A) and (B) is f andwhich is f ′.

Figure 2

33. Figure 3 shows the graph of position as a function of time. Determine the intervals onwhich the acceleration is positive.

10 20 30 40

time

position

Figure 3

Roughly from time 15 to time 20 and from time 32 on.34.

Which of the following descriptions could not apply to Figure 4.(a) The graph of acceleration for an object moving at constant velocity.(b) The graph of velocity for an object whose acceleration is constant.(c) The graph of position for an object whose acceleration is zero.

Figure 4

35. R & W In a bottle containing p fruit flies, the number of offspring produced by eachfemale is f (p) = (34 − 0.612p)p−0.658.(a) Calculate f ′′(10).(b) Give a verbal interpretation of f ′′(10).

Let f (p) = (34 − 0.612p)p−0.658 = 34p−0.658 − 0.612p0.342. Thenf ′(p) = 22.372p−1.658 − 0.209304p−0.658 andf ′′(p) = 37.092776p−2.658 + .137722032p−1.658.(a) Therefore, f (10) ≈ 6.1276, f ′(10) ≈ −.5377, and f ′′(10) ≈ 0.08455.(b) The quantity f (10) ≈ 6.1276 shows that at Day 10 each female fly is producing

roughly 6 offspring per day. The quantity f ′(10) ≈ −.5377 means that thereproduction rate is declining at Day 10. This shows that the flies are becoming lessfertile as their population increases (due to overcrowding in the bottle). Finally,f ′′(10) ≈ .08455 signifies that the rate at which this decline in reproduction rateoccurs is becoming less pronounced; i.e., less negative.

36.Find the second derivative of the volume of a cube with respect to the length of a side.



37. Show that for any three values A, B, C there exists a unique quadratic polynomial f (x) suchthat f (0) = A, f ′(0) = B, and f ′′(0) = C .Let f (x) = px2 + qx + r , where p, q , and r are constants. Then f ′(x) = 2px + q andf ′′(x) = 2p. Setting f (0) = A, f ′(0) = B, and f ′′(0) = C gives r = A, q = B, and2p = C . Hence p = C/2, q = B, and r = A.

38.Use the Product Rule twice to find a formula for ( f g)′′ in terms of the derivatives of f andg.39. Use the Product Rule to find a formula for ( f g)′′′ and compare your result with theexpansion of (a + b)3. Then try to guess the general formula for ( f g)(n).Continuing from Exercise ??, we have

h′′′ = f ′′g′ + g f ′′′ + 2( f ′g′′ + g′ f ′′) + f g′′′ + g′′ f ′ = f ′′′g + 3 f ′′g′ + 3 f ′g′′ + f g′′′

The binomial theorem gives(a + b)3 = a3 + 3a2b + 3ab2 + b3 = a3b0 + 3a2b1 + 3a1b2 + a0b3 and more generally(a + b)n =

n∑k=0

(n

k

)an−kbk , where the binomial coefficients are given by(

n

k

)= k(k − 1) · · · (k − n + 1)

n! . Accordingly, the general formula for ( f g)(n) is given by

( f g)(n) =n∑

k=0

(n

k

)f (n−k)gk , where p(k) is the kth derivative of p (or p itself when k = 0).

40.

In a chemical reaction such as 2N2O5 → 4NO2 + O2, the reaction rate is the derivativedC/dt , where C(t) is the concentration of N2O5 at time t . Experiments show that

dC

dt= −kC (1)

where k is a positive constant. The derivative dC/dt is negative because the concentrationC(t) is decreasing.

(a) Show that d2C/dt2 = k2C .(b) Is the reaction rate increasing or decreasing?(c) When does the reaction rate change most quickly: at the beginning when C is large or

toward the end when C is small?

3.6 Trigonometric Functions


1. Determine if the following equations are true or false and if false, state the correct version.

(a)d

dx(sin x + cos x) = cos x − sin x

(b)d

dxtan x = sec x

(c)d

dxcsc x = − cot x

2. Which of the following functions can be differentiated using the rules we have covered sofar?

(a) 3 cos x cot x

(b)d

dxcos(x2)

(c)d

dxx2 cos x

3. Compute d/dx(sin2 x + cos2 x) without using the derivative formulas for sin x and cos x .

4. How is the addition formula used in deriving the formula (sin x)′ = cos x?

3.6 Trigonometric Functions 35

Exercises

In Exercises 1–4, find the equation of the tangent line at the point indicated.

1. sin x; x = π4

Let f (x) = sin x . Then f ′(x) = cos x and f ′( π4) =

√2

2.

2.cos x; x = π

33. tan x; x = π4

Let f (x) = tan x . Then f ′(x) = sec2 x and f ′ ( π4

) = 2.4.

sec x; x = π6

In Exercises 5–26, use the Product and Quotient Rules as necessary to find the derivative of eachfunction.

5. sin x cos x

Let f (x) = sin x cos x . Then f ′(x) = − sin2 x + cos2 x .6.

x2 cos x7. sin2 x

Let f (x) = sin2 x = sin x sin x . Then f ′(x) = 2 sin x cos x .8.

9 sec x + 12 cot x9. x3 sin x

Let f (x) = x3 sin x . Then f ′(x) = x3 cos x + 3x2 sin x .10.

sin x

x11. tan x sec x

Let f (x) = tan x sec x . Thenf ′(x) = tan x sec x tan x + sec x sec2 x = sec x tan2 x + sec3 x or (tan2 x + sec2 x) sec x .

12. xcos x13. x2 sin2 x

Let f (x) = x2 sin2 x . Thenf ′(x) = x2 (2 sin x cos x) + 2x sin2 x = 2x2 sin x cos x + 2x sin x . (Here we used the resultfrom Exercise 7.)

14.cos2 x

15. (x − x2) cot xLet f (x) = (x − x2) cot x . Then f ′(x) = (x − x2)(− csc2 x) + cot x(1 − 2x).

16.z tan z

17.sec x

x2

Let f (x) = sec xx2

. Then f ′(x) = sec x tan x(x2) − 2x sec x

x4.

18. xcos x

19. sin t − 2cos t

Let f (t) = sin t − 2cos t

= sin t − 2 sec t . Then f ′(t) = cos t − 2 sec t tan t .20.

cos y − 1sin y


21.x

sin x + 2Let f (x) = x

2 + sin x . Then f′(x) = (2 + sin x) (1) − x cos x

(2 + sin x)2 =2 + sin x − x cos x

(2 + sin x)2 .22.

(x2 − x − 4) csc x23.

1 + sin x1 − sin xLet f (x) = 1 + sin x

1 − sin x . Thenf ′(x) = (1 − sin x) (cos x) − (1 + sin x) (− cos x)

(1 − sin x)2 =2 cos x

(1 − sin x)2 .24.

1 + tan x1 − tan x25. sec x

x

Let f (x) = sec xx

. Then f ′(x) = x sec x tan x − (sec x) (1)x2

= (x tan x − 1) sec xx2

.26.

sin x

4 + cos xIn Exercises 27–30, calculate the second derivative of the function.

27. 3 sin x + 4 cos xLet f (x) = 3 sin x + 4 cos x . Then f ′(x) = 3 cos x − 4 sin x andf ′′(x) = −3 sin x − 4 cos x .

28.cot x

29. tan x

Let f (x) = tan x . Then f ′(x) = sec2 x = sec x sec x andf ′′(x) = 2(sec x)(sec x tan x) = 2 sec2 x tan x .

30. csc x

In Exercises 31–36, find the equation of the tangent line at the point specified.

31. x2 + sin x ; x = 0Let f (x) = x2 + sin x . Then f ′(x) = 2x + cos x and f ′ (0) = 1. The tangent line at x = 0isy = f (0) + f ′ (0) (x − 0) = 0 + 1 (x − 0) or y = x .

32.x tan x ; x = π/4

33. 2 sin x + 3 cos x ; x = π/3Let f (x) = 2 sin x + 3 cos x . Then f ′(x) = 2 cos x − 3 sin x and f ′ ( π

3

) = 1 − 3√32

. Thetangent line at x = π3 isy = f ( π3 )+ f ′ ( π3 ) (x − π3 ) = √3 + 32 + (1 − 3√32 ) (x − π3 ) =(1 − 3

√3

2

)x + √3 + 32 +

√3

2 π − π3 .34.

x2 + sin x ; x = 035. csc x − cot x ; x = π/4

Let f (x) = csc x − cot x . Then f ′(x) = csc2 x − csc x cot x andf ′(

π

4

) = 2 − √2 · 1 = 2 − √2. Hence the tangent line isy = f ( π4 )+ f ′ ( π4 ) (x − π4 ) = (√2 − 1)+ (2 − √2) (x − π4 ). ory =

(2 − √2

)x + √2 − 1 + π4

(√2 − 2

).

36. cos x1 + sin x ; x = π/3

3.6 Trigonometric Functions 37

37. Use the formulas (sin x)′ = cos x and (cos x)′ = − sin x together with the rules ofdifferentiation to verify the formulas.

(a)d

dxcot x = − csc2 x

(b)d

dxsec x = sec x tan x

(c)d

dxcsc x = − csc x cot x

(a)d

dx(cot x) = d

dx

(cos xsin x

)= (sin x)(− sin x) − cos x cos x

sin2 x

= − 1sin2 x

= − csc2 x .

(b)d

dx(sec x) = d

dx

(1

cos x

)= −− sin x

cos2 x= sec x tan x .

(c)d

dx(csc x) = d

dx

(1

sin x

)= − cos x

sin2 x= − csc x cot x .

38.Find the values of x between 0 and 2π at which the tangent line to the graph ofy = sin x cos x is horizontal.39. Let f (x) = cos x .(a) Calculate the first five derivatives of f .(b) What is f (8)?(c) What is f (37)?

Let f (x) = cos x .(a) Then f ′(x) = − sin x , f ′′(x) = − cos x , f ′′′(x) = sin x , f (4)(x) = cos x , and

f (5)(x) = − sin x .(b) Accordingly, the successive derivatives of f cycle among

{− sin x, − cos x, sin x, cos x} in that order. Since 8 is a multiple of 4, we havef (8)(x) = cos x .

(c) Since 36 is a multiple of 4, we have f (36)(x) = cos x , whence f (37)(x) = − sin x .40.

Calculate f ′′(x) and f ′′′(x), where f (x) = tan x .41. Find y(157), where y = sin x .

Let f (x) = sin x . Then the successive derivatives of f cycle among{cos x, − sin x, − cos x, sin x} in that order. Since 156 is a multiple of 4, we havef (156)(x) = sin x , whence f (157)(x) = cos x .

42.

(a) Show that no tangent line to the graph of tan x has zero slope.(b) Among all tangent lines, which one has the least slope?(c) Justify your answer by sketching the graph of (tan x)′.

43. A weight attached to a spring is oscillating up and down. Its height above ground at time tseconds is s(t) = 300 + 40 sin t cm. Find the velocity and acceleration of the weight attime t = π

3seconds.

Let h(t) = 300 + 40 sin t be the height. Then the velocity is v(t) = h′(t) = 40 cos t andthe acceleration isa(t) = v′(t) = −40 sin t . At t = π3 , the velocity is v

(π

3

) = 20 and the acceleration isa(

π

3

) = −20√3.44.

GU Let g(t) = t − sin t .(a) Sketch the graph of g with a graphing utility for 0 ≤ t ≤ 4π .(b) Show that the slope of the tangent line is always positive and verify this on your graph.(c) For which values of t in the given range is the tangent line horizontal?



45. Prove that d cos x/dx = − sin x using the limit definition of the derivative and the additionlaw for the cosine: cos(a + b) = cos a cos b − sin a sin b.Let f (x) = cos x . Then

f ′(x) = limh→0

cos(x + h) − cos xh

= limh→0

cos x cos h − sin x sin h − cos xh

= limh→0

((− sin x) sin h

h+ (cos x) cos h − 1

h

)= (− sin x) · 1 + (cos x) · 0 = − sin x .

46.

R & W This exercise derives the formula (cos x)′ = − sin x from (sin x)′ = cos x usingthe identity

cos x = sin(

x + π2

)This identity says that the graph of cos x is obtained by shifting the graph of sin x to theright π/2 units.

(a) Explain in a few sentences why the derivative of cos x is obtained by shifting thederivative of sin x to the right π/2 units.

(b) Which function do you get when you shift the graph of cos x to the right π/2 units?(c) Use (a) and (b) to explain why (cos x)′ = − sin x .

47. This exercise proves that sin x is not a polynomial function.(a) Show that the function y = sin x satisfies y ′′ = −y.(b) Show that a nonzero polynomial cannot satisfy y ′′ = −y. Hint: if y is a polynomial,

then y ′′ is a polynomial of lower degree than y.(c) Does the same reasoning show that cos x is not a polynomial?

(a) Let y = sin x . Then y ′ = cos x and y ′′ = − sin x . Therefore, y ′′ = −y.(b) Let p be a nonzero polynomial of degree n and assume that p satisfies the differential

equation y ′′ + y = 0. Then p′′ + p = 0 for all x . There are exactly three cases.i. If n = 0, then p is a constant polynomial and thus p′′ = 0. Hence 0 = p′′ + p = p

or p ≡ 0 (i.e., p is equal to 0 for all x or p is identically 0). This is a contradiction,since p is a nonzero polynomial.

ii. If n = 1, then p is a linear polynomial and thus p′′ = 0. Once again, we have0 = p′′ + p = p or p ≡ 0, a contradiction since p is a nonzero polynomial.

iii. If n ≥ 2, then p is a quadratic polynomial and thus p′′ is a polynomial of degreen − 2 ≥ 0. Thus q = p′′ + p is a polynomial of degree n ≥ 2. By assumption,however, p′′ + p = 0. Thus q ≡ 0, a polynomial of degree 0. This is acontradiction, since the degree of q is n ≥ 2.

CONCLUSION: In all cases, we have reached a contradiction. Therefore theassumption that p satisfies the differential equation y ′′ + y = 0 is false. Accordingly, anonzero polynomial cannot satisfy the stated differential equation.

(c) If f (x) = cos x were a polynomial, then f (k), the kth derivative of cos x , would be 0for some k. This is a contradiction, since the successive derivatives of f cycle among{− sin x, − cos x, sin x, cos x}.

48.

This exercise gives an alternate derivation of the formula sin′ x = cos x using the additionformulas

sin(a + b) = sin a cos b + cos a sin bsin(a − b) = sin a cos b − cos a sin b

(a) Set x = a + b and y = a − b. Express the above identities in terms of x and y, andsubtract them to obtain

sin x − sin y = 2 cos(

x + y2

)sin

(x − y

2

)

(b) Conclude that

sin(x + h) − sin x = 2 cos (x + 12h)

sin(

12h)

(c) Use the identity in (b) to compute (sin x)′.

49. Use the addition formula for the sine function to show thattan(x + h) − tan x = sin h sec x sec(x + h). Then use this identity to verify thatd/dx tan x = sec2 x .The addition formula for sine is sin(a + b) = sin a cos b + sin b cos a and the additionformula for cosine is cos(a + b) = cos a cos b − sin a sin b. We rewrite tan(x + h) − tan x

3.7 The Chain Rule 39

as :

sin(x + h)cos(x + h) −

sin x

cos x

= sin x cos h + sin h cos xcos x cos h − sin x sin h −

sin x

cos x

= sin x cos h cos x + sin h cos2 x − sin x cos h cos x + sin2 x sin h

cos x(cos x cos h − sin x sin h)= sin h(cos

2 x + sin2 xcos x(cos x cos h − sin x sin h)

= sin hcos x(cos(x + h))

= sin h sec x sec(x + h)We verify that d

dxtan x = sec2 x by using the definition of limit:

d

dxtan x = lim

h→0tan(x + h) − tan x

h= lim

h→0sin h sec h sec(x + h)

h

= limh→0

sin h

h· lim

h→0sec x sec(x + h) = sec2 x

50.

Let f (x) = x sin x and g(x) = x cos x .(a) Show that f ′(x) = g(x) + sin x and g′(x) = − f (x) + cos x .(b) Differentiate the formulas in (a) to verify that f ′′(x) = − f (x) + 2 cos x and

g′′(x) = −g(x) − 2 sin x .(c) Try, by further experimentation, to find formulas for all higher derivatives of both f

and g. Hint: the expression for the kth derivative will depend on whetherk = 4n, 4n + 1, 4n + 2, or 4n + 3.

51. (Exercises 51 and 52 are taken from Trigonometry, by I.M. Gelfand and M. Saul,Birkhauser, 2001.) Show that if π/2 < θ < π , then the distance along the x-axis between θand the point where the tangent line intersects the x-axis is equal to | tan θ |.

πθ

sin x

2π

Figure 1

Let f (x) = sin x . Since f ′(x) = cos x , this means that the tangent line at (θ, sin θ) isy = cos θ(x − θ) + sin θ . When y = 0, x = θ − tan θ . Distance from x to θ is:

|θ − (θ − tan θ)| = | tan θ |

52.What is the distance along the x-axis between θ and the point where the tangent lineintersects the x-axis, if 0 < θ < π/2?

3.7 The Chain Rule


1. Identify the outside and inside functions for each of the composite functions.

Documents

3.1 Deﬁnition of the Derivativedeturck/m103/rogawski/chap3...3.1 Deﬁnition of the Derivative 3 In Exercises 6–9, compute the derivative of the quadratic polynomial at the point