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146 Elementos de cálculo, volumen 1
29. 2x
−2
31.
(x + 3) 2
33.
−2x
(x 2 + 1) 2
35. f ′ (2) ≈ 1, 2
3
37.
2 √ 5
+
x 3 x−2/3
39. −t4 − t 3 + 9t 2 + 9t
(t 2 + 3t) 2
41. −3t2 − 8t √ t − 3
2 √ t(t 2 − 3) 2
43. −3x −2 − 5
2 x−7/2 +6x −3 −24x 3 −
5x 3/2 + 18x 2
45. − 2
5 (2x)−4/5 (x −3 + x 2 − x)+
(−3x −4 +2x−1)(4− 5√ 2x)
47. −20x4/3 − 4x1/3 − 6x2 + 12x − 3
(2x2 + x) 2
49.
3
2 √ 3t + 1
51.
−4x − 1
(3x + 2) 2 √ 2x + 1
53. (x3 − 2x) 4 (−x 6 + 9x 4 + 6x 2 )
(x 4 + 3x 2 ) 6
55. −25t −6 + 1
4 t−3/4 + 1
2 t−1/2
57. 6x4 + 16x 3 + 17x 2 + 8x + 12
(2x + 1) 2
59. 2 + 5(3x2 + x)
√ 2x 3 + x 2
61. f ′ (x) = 3x2 − 2, f ′′ (x) = 6x,
f ′′′ (x) = 6, f IV (x) = 0
63. h ′ (x) = 1
√ , h
x ′′ (x) = −1
2 √ x3 , h′′′ (x) = 3
4 √ x
h IV (x) = −15
8 √ x7 65. g ′ (r) = −15
2 r−7/2 + 2
3 r−2/3 ,
g ′′ (r) = 105
4 r−9/2 − 4
9 r−5/3 ,
g ′′′ (r) = −945
8 r−11/2 + 20
27 r−8/3 ,
g IV (r) = 10395
16 r−13/2 − 160
81 r−11/3
67. y ′ =
2x − y
x + 3y
69. y ′ = −3x2 y 2 − y
2x 3 y + x − 2
71. v = 20 m/seg
73. y = x
75. y = −5x − 7
77. Recta tangente: y = −
√ 2
5 ,
√
2
6 x +
3 2 , recta normal: y = 3√2 x+3−
√
2
5
3
79. (−1, 13) y (2, 26)
81. (− 1
2 , −1) y (2, 9)
83. (1, 2) y (−1, 2)
85. c = − 1
4