Chapter 3
Section 3.1
1. Chain
2. True
3. False
4. tan u; 1 + cos x
5. 100(x3 + 4x + 1)99 (3x2 + 4)
6. 6xe3x2+5
7. True
8. 2x cos(x2)
9. y = (x3 + 1)5; 
= 15x2(x3 + 1)4
11. 
13. 
15. f′(x) = 6(3x + 5)
17. f′(x) = 
19. g′(x) = 8x(x2 + 5)3
21. f′(u) = 
23. g′(x) = 3(4x + ex)2(4 + ex)
25. f′(x) = 2 tan x sec2 x
27. f′(z) = 2(tan z +cos z)(sec2 z −sin z)
29. y′ = 4x(x2 +4)(2x3 −1)3 +18x2(x2 +4)2(2x3 −1)2 = 2x(x2 +4)(2x3 −1)2(13x3 +36x −2)
31. y′ = 
33. y′ = 4 cos(4x)
35. y′ = 2 cos(x2 + 2x − 1) · (2x + 2)
37. y′ = 
39. y′ = 4 sec(4x) tan(4x)
41. y′ = e1/x
43. y′ = 
45. y′ = 
47. y′ = (ln 2)2sin x cos x
49. y′ = (ln 6)6sec x sec x tan x
51. y′ = 5e3x + 15xe3x
53. y′ = 2x sin(4x) + 4x2 cos(4x)
55. y′ = −ae−ax sin(bx) + be−ax cos(bx)
57. y′ = 
59. 
61. 
63. y′ = −2e−2x cos(3x) − 3e−2x sin(3x)
65. y′ = −2xex2 sin(ex2)
67. y′ = ecos(4x)(−4 sin(4x))
69. y′ = 24 sin(3x) cos(3x)
71. (a) y′ = 6x2(x3 + 1)
(b) y′ = 6x2(x3 + 1)
(c) y′ = 6x5 + 6x2
73. (a) y = 0
(b)
75. (a) y = 
(b)
77. (a) y = 2x + 1
(b)
79. 
= −25x8 cos(x5) − 20x3 sin(x5)
81. h′(1) = −12
83. h′(0) = 0
85. 78
87. 
= f′(x2 + 1)(2x)
89. 
91. 
= f′(sin x) cos x
93. 
= f″;(cos x) sin2 x − f′(cos x) cos x
95. (a) 
(b) 
(c) 
(d) 
97. a(t) = 
99. 
= −3.597 × 10−6 ohms/K
101. (a) 102
(b) 
(c) 
= 18.9e−0.21t
(d) A′(5) = 6.614 %/yr; A′(10) = 2.314 %/yr
(e)
103. (a) a(t) = 
(b) mg/k
(c) 0
105. See Student Solutions Manual.
107. (a) 
(b) 
109. F′(1) = 2
111. See Student Solutions Manual.
113. See Student Solutions Manual.
115. See Student Solutions Manual.
117. See Student Solutions Manual.
119. See Student Solutions Manual.
121. (a) y(n)(x) = aneax
(b) y(n)(x) = (−1)nane−ax
123. (a)
= a11 sin(ax)
(b) 
= a12 cos(ax)
(c) f(n)(x) = −an sin(ax), n = 1 + 4k, k = 0, 1, 2, 3,…; f(n)(x) = −an cos(ax), n = 2 + 4k, k = 0, 1, 2, 3,…; f(n)(x) = an sin(ax), n = 3 + 4k, k = 0, 1, 2, 3,…; f(n)(x) = an cos(ax), n = 4 + 4k, k = 0, 1, 2, 3,…
125. See Student Solutions Manual.
127. See Student Solutions Manual.
129. See Student Solutions Manual.
131. See Student Solutions Manual.
133. (a) −72,000π2 cm/sec2
(b) 72,000π2 kg · cm/sec2
Section 3.2
1. True
2. True
3. True
4. 
5. False
6. 
7. y′ = 
9. y′ = 
11. 
13. y′ = 
15. y′ = 
17. y′ = 
19. y′ = 
21. y′ = 
23. 
25. y′ = 
27. y′ = 
29. y′ = 
31. y′ = 
33. y′ = 
35. y′ = 
37. y′ = 
39. y′ = (x2 − 1)1/2 + 
41. y′ = 
43. y′ = 
(x2 cos x)1/2(2x cos x − x2 sin x)
45. y′ = 3x(x2 − 3)1/2(6x + 1)5/3 + 10(x2 − 3)3/2(6x + 1)2/3 = (6x + 1)2/3(x2 − 3)1/2(28x2 + 3x − 30)
47. y′ = 
49. y′ = 
51. y′ = 
53. y′ = sin−1 x + 
55. y′ = 
57. 
59. y′ = 
61. y′ = 
63. (a) −1/2
(b) y = 
(c)
65. (a) 3
(b) y = 3x – 8
(c)
67. (a) 1/2
(b) y = 
(c)
69. y′ = −4/9; y″; = −100/243
71. g′(4) = −1/2
73. f′(−2) = 2
75. g′(0) = 1/2, g′(3) = 1/5
77. y′ = 3x/|x|
79. y′ = 
81. 
83. 
85. y = −2x + 10
87. (a) 
(b) 
89. (a) y′ = 
(b) y = 
(c) (6,−3)
(d)
91. (a) y = 
(b)
93. See Student Solutions Manual.
95. See Student Solutions Manual.
97. (a) y′ = 
(b) y = −x + 2
(c) 
(d)
99. 
101. See Student Solutions Manual.
103. See Student Solutions Manual.
105. See Student Solutions Manual.
107. See Student Solutions Manual.
109. See Student Solutions Manual.
111. (a) 
(b)
113. See Student Solutions Manual.
115. See Student Solutions Manual.
Section 3.3
1. 
2. True
3. False
4. False
5.
6. e
7. y′ = 5/x
9. y′ = 
11. y′ = 
13. y′ = 1/x
15. y′ = 
17. y′ = ln(x2 + 4) + 
19. y′ = 
21. y′ = 
23. y′ = 
25. y′ = 
27. y′ = 
29. y′ = cot θ
31. y′ = 
33. y′ = 
35. y′ = 
37. y′ = 
39. y′ = 
41. y′ = cos(ln θ) · 
43. y′ = 
45. 
47. 
49. 
51. y′ = 
53. y′ = 
55. y′ = 
57. y′ = (3x x (ln(3x) + 1)
59. y′ = 
61. y′ = xx2 (2x ln x + x)
63. y′ = 
65. y′ = 
67. y′ = (sin x)x(ln(sin x) + x cot x)
69. y′ = (sin x)cos x (−sin x ln(sin x) + cos x cot x)
71. y′ = 
73. y = 5x − 1
75. y = −5x + 18
77. e2
79. e1/3
81. 
83. 
85. y′ = xx(ln x + 1)
87. y′ = tan−1(x/a)
89. (a) See Student Solutions Manual.
(b) $6107.01
(c) 28.881years
(d) See Student Solutions Manual.
91. (a) 
dB/m
(b) Answers will vary.
93. See Student Solutions Manual.
95. See Student Solutions Manual.
97. See Student Solutions Manual.
Section 3.4
1. (d)
2. f (x0) + f ′(x0)(x − x0)
3. True
4. 
5. True
6. True
7. dy = (3x2 − 2) dx
9. dy = 12x(x2 + 1)1/2 dx
11. dy = (6 cos(2x) + 1) dx
13. dy = −e−x dx
15. dy = (ex + xex)dx
17. (a) dy = exdx
(b) (i) 
(ii) 
(iii) 
(c) (i) 0.4043 (ii) 0.0141 (iii) 0.000136
19. (a) 
(b) (i) dy ≈ 0.2646, Δy ≈ 0.2546 (ii) dy ≈ 0.0529, Δy ≈ 0.0525 (iii) dy ≈ 0.0053, Δy ≈ 0.0053;
(c) (i) 0.009952, (ii) 0.000431, (iii) 4 × 10−6
21. (a) dy = −sin x dx
(b) (i) dy = 0, Δy ≈ 0.1224 (ii) dy = 0, Δy ≈ 0.005 (iii) dy = 0, Δy ≈ 0.00005;
(c) (i) 0.1224, (ii) 0.005 (iii) 0.00005
23. (a) L(x) = 405x – 567
(b)
25. (a) L(x) = 
(b)
27. (a) L(x) = x – 1
(b)
29. (a) L(x) = 
(b)
31. (a) 0.006
(b) 0.00125
33. (a) See Student Solutions Manual.
(b) c3 = 1.155
35. (a) See Student Solutions Manual.
(b) c3 = 0.214
37. (a) See Student Solutions Manual.
(b) c3 = 3.136
39. (a) See Student Solutions Manual.
(b) Answers will vary.
41. (a) See Student Solutions Manual.
(b) c5 ≈ −0.567
43. (a) See Student Solutions Manual.
(b) c5 ≈ −0.791
45. (a) See Student Solutions Manual.
(b) c5 ≈ 4.796
47. 2π
49. 3.6π m3
51. (a) 1.6π m3
(b) Underestimate
(c) Answers will vary.
53. 30 ± 0.27m; 0.9%
55. 72 min
57. 9.998 kg
59. 0.333 %
61. 6%
63. 1.309
65. -0.703
67. 2.718
69. Answers will vary.
71. See Student Solutions Manual.
73. See Student Solutions Manual.
75. See Student Solutions Manual.
77. 9.871 in
Section 3.5
1. False
2. True
3. P5(x) = 3(x − 1)3 + 11(x − 1)2 + 7(x − 1) + 4
5. P5(x) = 2(x + 1)4 − 14(x + 1)3 + 30(x + 1)2 − 25(x + 1) + 7
7. P5(x) = (x−2)5+10(x−2)4+40(x−2)3+80(x−2)2+80(x−2)+32
9. P5(x) = 
11. P5(x) = −(x −1)5+(x −1)4−(x −1)3+(x −1)2−(x −1)+1
13. P5(x) = 
15. P5(x) = 
17. P5(x) = x5 + x4 + x3 + x2 + x + 1
19. P5(x) = −6x5 + 5x4 − 4x3 + 3x2 − 2x + 1
21. P5(x) = 
23. P5(x) = 
25. P5(x) = 
27. P5(x) = 
29. f (x) = 3(x − 1)2 + 1
31. f (x) = (x − 1)3 + 4(x − 1)2 + 5(x − 1) − 6
33. (a) P4(x) = 
(b)
35. (a) P5(x) = 
(b)
37. (a) P4(x) 
(b)
39. (a) P3(x) = 3.205 + 0.620x + 0.058x2 + 0.0034x3
(b)
41. See Student Solutions Manual.
43. P2 (x) = − λπ − (1 + λ)(x − π) See Student Solutions Manual.
Section 3.6
1. False
2. sinh x/ cosh x
3. False
4. (a)
5. False
6. False
7. True
8. False
9. 
11. 1
13. 0
15. See Student Solutions Manual.
17. See Student Solutions Manual.
19. See Student Solutions Manual.
21. See Student Solutions Manual.
23. See Student Solutions Manual.
25. y′ = 3 cosh(3x)
27. y′ = 2x sinh(x2 + 1)
29. y′ = 
31. y′ = 4 sinh x sinh(4x) + cosh x cosh(4x)
33. y′ = 2 cosh x sinh x
35. y′ = ex sinh x + ex cosh x
37. y′ = 2x sech x − x2 tanh x sech x
39. y′ = 
41. y′ = 
43. 
45. 
47. y′ = 
49. P6(x) = 
51. Δ ≈ 0.031 radians ≈ 1.776°
53. (a) 625.092 ft
(b) 580.744 ft
(c) ±0.384
(d) ±2.766
(e) 82.026°
(f)
55. (a) See Student Solutions Manual.
(b) { x | x > 1 }
57. See Student Solutions Manual.
59. See Student Solutions Manual.
61. See Student Solutions Manual.
63. See Student Solutions Manual.
65. 0
67. See Student Solutions Manual.
Review Exercises
1. an(ax + b)n−1
3. 
5. 3x(x2 + 4) 1/2
7. 
9. 
11. 
13. sec(2x) + 2x sec(2x) tan(2x)
15. 
17. cos v − sin2 v cos v = cos3 v
19. 1.05x ln 1.05
21. 
23. 2 cot(2x)
25. 
27. 
29. 
31. 
33. 
35. 
37. tan−1(x)
39. 
41. 
43. 
45. 
47. 
49. y′ = 
51. y′ = 
53. 
55. e2/5
57. 0
59. See Student Solutions Manual.
61. 
63. (a) 
(b) 
(c) 
(d) 
65. dy = 
67. L(x) = 2x − 1
69. (a) dy = dx; Δy = tan(Δx)
(b) (i) dy = 0.5, Δy ≈ 0.546; (ii) dy = 0.1, Δy ≈ 0.100; (iii) dy = 0.01, Δy ≈ 0.010
71. 
73. −2
75. P4(x) = 
77. P6(x) = 
79. (a) See Student Solutions Manual.
(b) 2.568
81. y = 6x − 3