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C H A P T E R 2 Differentiation
Section 2.1 Section 2.2 Section 2.3 Section 2.4 Section 2.5 Section 2.6 The Derivative and the Tangent Line Problem . . . 53 Basic Differentiation Rules and Rates of Change . 60 The Product and Quotient Rules and Higher-Order Derivatives . . . . . . . . . . . . . . 67 The Chain Rule . . . . . . . . . . . . . . . . . . . 73 Implicit Differentiation . . . . . . . .. . . . . . . 79 Related Rates . . . . . . . . . . . . . . . . . . . . 85 . . . . . . . . . . . . . . . . . . . . . . . . . 92

Review Exercises

Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . 98

C H A P T E R Differentiation
Section 2.1

2

The Derivative and the Tangent Line Problem

Solutions to Odd-Numbered Exercises

1. (a) m (b) m

0 3

3. (a), (b)(c) y

f 4 4 3 x 3 1x

f 1 x 1 1 1 2 2

1

f 1)

y
y
6 5 4 3 2 1

f )4) 4

f )1) )x 1

1)

f )1)

x

1

f )4)

5 )4, 5)

x
f )4) f )1)
3

1

f )1) )1, 2)

2

x
1 2 3 4 5 6

5. f x

3

2x is a line. Slope

2

7. Slope at 1,

3

lim lim

g1 1 1 x

x→0

x x
2

g1 4 x x x 2
2

3 1

x→0

lim

2 x 2 x

x→0

lim 2
x→0

9.Slope at 0, 0

lim lim

f 0 3 t

t t t t t 3

f 0
2

t→0

11. f x 0 f x

3 lim f x 3 x 0 3
x→0

t→0

x x

f x

lim 3
t→0

lim

x→0

lim 0
x→0

13. f x f x

5x lim f x 5x 5 5
x→0

15. h s x x x x f x h s 5x

3 lim

2 s 3 hs 2 s 3
s→0

s s

hs 2 s 3

lim lim

x→0

3 lim
s→0

s s

3

x→0

2 s 3 lim s→0 s

2 3

53

54

Chapter 2Differentiation

17. f x f x

2x2 lim lim
x→0

x f x 2x 2x2 4x x

1 x x x
2

f x x 2 x
2

x x
2

1 x x x

2x2 1

x

1 2x2 x 1

x→0

lim lim 19. f x f x x3

4x x 2 x x

x→0

x

x→0

lim 4x
x→0

2 x

1

4x

1

12x f x x x3 x x x
3

lim lim lim

f x 12 x x 3x x
2 2

x→0

x x x
3

x3
3

12x 12x 12 x x3 12x

x→0

3x2 x

x→0

x3x2 x 3x x 3x x 12 x 3x2 12 x x
2

lim

x→0

lim 3x2
x→0

12

21. f x f x

1 x lim
x→0

1 f x 1 x x x 1 x x x x x 1
2

f x 1 x 1

lim

x

x→0

lim lim lim

x xx xx x 1

1

x→0

x 1 x x 1 x

1 1 1

x→0

x→0

1 1 x

1

x

23. f x f x

x lim
x→0

1 f x x x x x 1 1 x x x x x x x x 1 1 1 f x 1 x 1 1 x 1 2 x x x x 1 1 1 1 1 x x x x 1 1 x x 1 1

limlim lim

x→0

x→0

x→0

x

Section 2.1

The Derivative and the Tangent Line Problem

55

25. (a) f x f x

x2 lim
x→0

1 f x x x x
2

17. (b) f x
−5

8

(2, 5)
5 −2

lim

x 2x x

x→0

1 x x
2

x2

1

lim

x→0

x x 2x

lim 2x
x→0

At 2, 5 , the slope of the tangent line is m 22 4. The equation of the tangent line is y y 5 5 y 4x 4x 4x x3 lim f xx 3x2 x
x→0

2 8 3.

27. (a) f x f x

18. (b) x x x3 x f x
−5

10

(2, 8)

5 −4

lim lim

x3
2

x→0

x→0

3x x x 3x x

x x
2

3

lim 3x2
x→0

3x2 32
2

At 2, 8 , the slope of the tangent is m The equation of the tangent line is y 8 y 29. (a) f x f x x lim lim lim f x x x x x x 1 x
x→0

12.

12 x 12x

2 16. 18. (b) x x x x x x x f x
−1

3

(1, 1)5 −1

x x x

x→0

x x

x x

x x

x→0

lim

x→0

1 2 x

At 1, 1 , the slope of the tangent line is m 1 2 1 1 . 2

The equation of the tangent line is y 1 y 1 x 2 1 x 2 1 1 . 2

56

Chapter 2

Differentiation

31. (a) f x f x

4 x lim f x
x→0

(b) x x x f x 4 x x xx x3 x x 2x 2 x
2

10

(4, 5)
− 12 12

x lim lim lim lim lim x2 x2
x→0

x x

x

4 x x4x 4 x x

−6

x→0

4x x 2 x x x x x x3 x

x→0

x x2 x x x 4 x x

x2 x

x x x2 x x→0 x x x x2 xx 4 1 x x x 4 x2 4
x→0

At 4, 5 , the slope of the tangent line is m 1 4 16 3 4

The equation of the tangent line is y y 5 3 x 4 3 x 4 2 3x2. Since the 35. Using the limit definition of derivative, f x 1 2x x .
1 2,

4

33. From Exercise 27 we know that f x slope of the given...