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CHAPTER 2
Differentiation
Section 2.1 The Derivative and the Tangent Line Problem . . . . . 95
Section 2.2 Basic Differentiation Rules and Rates of Change . . 109
Section 2.3 Product and Quotient Rules and
Higher-Order Derivatives . . . . . . . . . . . . . . . 120
Section 2.4 The Chain Rule . . . . . . . . . . . . . . . . . . . . 134
Section 2.5 Implicit Differentiation . . . . . . . . . . . . . . . . 147
Section 2.6 Related Rates . . . . . . . . . . . . . . . . . . . . . 161
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
CHAPTER 2
Differentiation
Section 2.1 The Derivative and the Tangent Line Problem
1. (a) At slopex , y , 0. 2. (a) At slopex , y , 2.
1 1 1 1 3
At slopex , y , 5. At slopex , y , 2.
2 2 2 2 2 5
(b) At slopex , y , 5. (b) At slopex , y , 4.
1 1 2 1 1 3
At slopex , y , 2. At slopex , y , 5.
2 2 2 2 4
3. (a), (b) 4. (a) f 4 f1 5 2 1
y f ) 4) f 1)) )x 1) f 1) ) x 1 4 1 3
4 1
y f 4 f3 5 4.75
6 4 3 1 0.25
5 f ) 45)
4 4) ,)5 Thus, f4 f1 > f4 f3.
f ) 4) f 13) ) 4 1 4 3
3
2 f ) 1) 2
1) , 2) (b) The slope of the tangent line at 1, 2 equals f1.
1 This slope is steeper than the slope of the line through
x
1 2 3 4 5 6 and Thus,
1, 2 4, 5.
(c) y f4 f1 x 1 f1) f 4 f1 < f1.
4 1 4 1
3x 1 2
3
1x 1 2
x 1
5. fx 3 2x is a line. Slope 2 6. gx 3x 1 is a line. Slope 3
2 2
7. Slope at 1, 3 lim g1 x g1 8. Slope at 2, 1 lim g2 x g2
x→0 x x→0 x
2 2
lim 1 x 4 3 lim 5 2 x 1
x→0 x x→0 x
2 2
lim 1 2x x 1 lim 5 4 4x x 1
x→0 x x→0 x
lim 2 x 2 lim 4 x 4
x→0 x→0
9. Slope at 0, 0 lim f0 t f0 10. Slope at 2, 7 lim h2 t h2
t→0 t t→0 t
2 2
lim 3t t 0 lim 2 t 3 7
t→0 t t→0 t
2
lim 3 t 3 4 4t t 4
t→0 lim
t→0 t
lim 4 t 4
t→0
95
96 Chapter 2 Differentiation
11. f x 3 12. gx 5 13. fx 5x
fx lim fx x fx gx lim gx x gx fx lim fx x fx
x→0 x x→0 x x→0 x
lim 3 3 lim 5 5 lim 5x x 5x
x→0 x x→0 x x→0 x
lim 0 0 0 lim 5 5
x→0 lim 0 x→0
x→0 x
14. fx 3x 2 15. hs 3 2s
3
fx lim fx x fx hs s hs
x→0 x hs lim
s→0 s
lim 3x x 2 3x 2 3 2s s 3 2s
x→0 x lim 3 3
s→0 s
3x 2 s 2
lim lim 3
x→0 x s→0 s 3
lim 3 3
x→0
16. fx 9 1x
2
fx lim fx x fx
x→0 x
lim 9 12x x 9 12x
x→0 x
1 1
lim
x→0 2 2
17. fx 2x2 x 1
fx lim fx x fx
x→0 x
2 2
lim 2x x x x 1 2x x 1
x→0 x
2 2 2
lim 2x 4x x 2x x x 1 2x x 1
x→0 x
2
lim 4x x 2x x lim 4x 2 x 1 4x 1
x→0 x x→0
18. fx 1 x2
fx lim fx x fx
x→0 x
2 2
lim 1 x x 1 x
x→0 x
2 2 2
lim 1 x 2x x x 1 x
x→0 x
2
lim 2x x x lim 2x x 2x
x→0 x x→0
Section 2.1 The Derivative and the Tangent Line Problem 97
19. fx x3 12x
fx lim fx x fx
x→0 x
3 3
lim x x 12x x x 12x
x→0 x
3 2 2 3 3
lim x 3x x 3xx x 12x 12 x x 12x
x→0 x
2 2 3
lim 3x x 3xx x 12 x
x→0 x
2 2 2
lim 3x 3x x x 12 3x 12
x→0
20. fx x3 x2
fx lim fx x fx
x→0 x
3 2 3 2
lim x x x x x x
x→0 x
3 2 2 3 2 2 3 2
lim x 3x x 3xx x x 2x x x x x
x→0 x
2 2 3 2
lim 3x x 3xx x 2x x x
x→0 x
2 2 2
lim 3x 3x x x 2x x 3x 2x
x→0
21. fx 1 22. fx 1
x 1 2
x
f x x fx fx lim fx x fx
fx lim x→0 x
x→0 x
1 1 1 1
2 2
x x1x1 lim x x x
lim x→0 x
x→0 x
2 2
lim x x x
x 1 x x 1 x→0 2 2
lim xx x x
x→0 xx x 1x 1 2
lim 2x x x
x x→0 2 2
lim xx x x
x→0 xx x 1x 1 2xx
lim 2 2
lim 1 x→0 x x x
x→0 x x 1x 1 2x
1 x4
2
x 1 2
3
x
23. fx x 1
fx lim fx x fx
x→0 x
lim x x1 x1 xx1 x1
x→0 x
x x1 x1
lim x x 1 x 1
x→0
x x x1 x1
lim 1 1 1
x→0
x x1 x1 x 1 x 1 2 x1
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