Lista de 100 derivadas con respuesta

Derivadas
c 2007-2010
MathCon Contenido
1. Derivadas
1.1. Derivadas directas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2. Derivadas implı́citas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
6
1
Derivadas
1.1. Derivadas directas
1. f (x) = x2 + x + 1
2. f (x) = xa + x
3. f (x) = (x2 + x)(x)
4. f (x) = (x + 1)(x + 2)
5. f (x) = (x2 − 1)(x3 + 2)
6. f (x) = (x2 + x + 1)(x3 − 2x + 3)
7. f (x) = sin2 (x)
8. f (x) = x cos(x)
√
9. f (x) = ( x)(x2 + 3)
10. f (x) =
√
3
x2 + a
√
11. f (x) = a x + a
R. f ′ (x) = 2x + 1.
R. f ′ (x) = axa−1 + 1.
R. f ′ (x) = 3x2 + 2x.
R. f ′ (x) = 2x + 3.
R. f ′ (x) = 5x4 − 3x2 + 4x.
R. f ′ (x) = 5x4 + 4x3 − 3x2 + 2x + 1.
R. f ′ (x) = sin(2x).
R. f ′ (x) = cos x − x sin x.
R. f ′ (x) =
R. f ′ (x) =
3 + 5x2
√ .
2 x
2x
.
3(a + x2 )( 2/3)
a
.
R. f ′ (x) = √
2 a+x
√
12. f (x) = (a2 − x2 ) x2 + a2
x(a2 + 3x2 )
.
R. f ′ (x) = − √
a2 + x2
√
√
x2 + b2 x2 + a2
x(a2 + b2 + 2x2 )
√
R. f ′ (x) = √
.
a2 + x2 b2 + x2
√
√
14. f (x) = ( x + 1)( x − 2)
1
R. f ′ (x) = 1 − √ .
2 x
√
√
15. f (x) = ( x + x)( x − x)
R. f ′ (x) = 1 − 2x.
13. f (x) =
1.1. DERIVADAS DIRECTAS
√
√
16. f (x) = ( x + x)/( x − x)
√
17. f (x) = 1/( x + 1)
√
18. f (x) = ( x + 1)/(x2 − x)
19. f (x) = x/ sin x
20. f (x) =
sin x
1 − cos x
21. f (x) = sin x(sin x + cos x)
1
√ .
R. f ′ (x) = √
( x − 1)2 x
1
√
√ .
2( x + 1)2 x
√
2−3 x
′
.
R. f (x) = √
2( x − 1)2 x2
R. f ′ (x) =
R. f ′ (x) = (1 − x cot x)(csc(x)).
1
.
cos x − 1
R. f ′ (x) =
R. f ′ (x) = cos 2x + sin 2x.
22. f (x) =
sec x
x
R. f ′ (x) =
sec x(x tan x − 1)
.
x2
23. f (x) =
x
1 − sin x
R. f ′ (x) =
1 + x cos x − sin x
.
(sin x − 1)2
24. f (x) =
x
a + bx2
√
25. f (x) = 1/ x
R. f ′ (x) =
R. f ′ (x) = −
√
26. f (x) = x/ x
√
27. f (x) = 1/ x + 1
a − bx2
.
(a + bx2 )2
1
√ .
2x x
1
R. f ′ (x) = √ .
2 x
R. f ′ (x) = −
1
.
2(1 + x)3/2
√
28. f (x) = x/ x2 + 1
R. f ′ (x) =
1
.
(1 + x2 )3/2
29. f (x) = x2 /(x3 + 1)
R. f ′ (x) = −
30. f (x) = (a − x)/(a + x)
R. f ′ (x) = −
1
33. f (x) = sin( )
x
34. f (x) = 1/ sin x
35. f (x) = 1/ sin(1/x)
2a
.
(a + x)2
R. f ′ (x) = cos(2x).
31. f (x) = sin x cos x
32. f (x) = ex (sin x + cos x)
x(x3 − 1)
.
(1 + x3 )2
R. f ′ (x) = 2ex cos x.
R. f ′ (x) = −
cos(1/x)
.
x2
R. f ′ (x) = − cot x csc x.
R. f ′ (x) =
cot(1/x) csc(1/x)
.
x2
36. f (x) = 1/ ln x
R. f ′ (x) = −
1
.
x ln2 (x)
37. f (x) = x/ ln x
R. f ′ (x) =
ln x − 1
.
ln2 (x)
1.1. DERIVADAS DIRECTAS
R. f ′ (x) =
38. f (x) = ln x/x
39. f (x) = x/(sin2 (x) + x)
R. f ′ (x) =
sin(x)(sin(x) − 2x cos(x))
.
(sin(x)2 + x)2
R. f ′ (x) = tan2 x.
40. f (x) = tan x − x
41. f (x) = (sin(x) − cos(x))/x
42. f (x) =
43. f (x) =
xm
xp
− am
√
x2 + a2
p
√
44. f (x) = x + x
45. f (x) =
r
46. f (x) =
√
a
ax + √
ax
x+1
x
R. f ′ (x) =
(x + 1) cos(x) + (x − 1) sin(x)
.
x2
R. f ′ (x) =
xp−1 (p − m)xm − pam
.
(xm − am )2
R. f ′ (x) =
1
R. f ′ (x) = − p
.
2 1 + 1/xx
a
a
R. f ′ (x) = √ − √ .
2 ax 2x ax
R. f ′ (x) = ln x + 1.
√
48. f (x) = ln( x)
√
49. f (x) = ln(1/ x)
r
x+1
x−1
R. f ′ (x) = −
r
a−x
a+x
R. f ′ (x) = −
53. f (x) =
r
a2 + x2
a2 − x2
R. f ′ (x) =
54. f (x) =
s
sin(x)
cos(x)
55. f (x) = sin(x2 ) cos2 (x)
√
56. f (x) = sin( x)/x
R. f ′ (x) =
1
.
2x
R. f ′ (x) = −
1
.
2x
R. f ′ (x) =
50. f (x) = ln(ln x)
52. f (x) =
x
.
x2 + a2
1
1+ √
2 x
R. f ′ (x) = p√
.
2
x+x
47. f (x) = x ln x
51. f (x) =
1 − ln x
.
x2
(x −
1
r
1)2
a
r
(a + x)2
1
.
x ln x
x+1
x−1
a−x
a+x
.
.
2a2 x
√
.
(a2 − x2 ) a4 − x4
sec2 (x)
R. f ′ (x) = p
.
2 tan(x)
R. f ′ (x) = 2 cos2x (x) cos(x2 ).
√
√
√
x cos( x) − 2 sin( x)
.
R. f ′ (x) =
2x2
1.1. DERIVADAS DIRECTAS
√
57. f (x) = e
58. f (x) =
√
e x
R. f (x) = √ .
2 x
x
′
ex − e−x
2
R. f ′ (x) =
ex + e−x
.
2
59. f (x) = ln(1/x)
R. f ′ (x) = −1/x.
60. f (x) = ln x ln x
R. f ′ (x) =
61. f (x) = ln(ln(ln(1/x)))
R. f ′ (x) = −
63. f (x) = ee
1
.
− x2
x
+x
.
R. f ′ (x) = xx (1 + ln x) .
65. f (x) = xx
x
66. f (x) = 23
x
67. f (x) = 23
5x
68. f (x) = ex ln x
R. f ′ (x) = xx
x
+x−1
(1 + x ln x + x ln2 (x)) .
R. f ′ (x) = 23 3x ln 2 ln 3 .
x
5x
R. f ′ (x) = 23 35 5x ln 2 ln 3 ln 5 .
70. f (x) = e1/x ex
a + bx
71. f (x) = e a − bx
72. f (x) = esin x
73. f (x) = ex sin x
74. f (x) = atan nx
75. f (x) = x1/x
x
77. f (x) = sinx x
x
R. f ′ (x) =
69. f (x) = e1/x
√
a2
R. f ′ (x) = ee
x
64. f (x) = xx
76. f (x) = x
1
.
x ln(1/x) ln(ln(1/x))
R. f ′ (x) = √
62. f (x) = arcsin(x/a)
2 ln x
.
x
ex (1 + x ln x)
.
x
R. f ′ (x) = −
e1/x
.
x2
R. f ′ (x) = e1/x+x (1 −
1
).
x2
a + bx
2abe a − bx
R. f ′ (x) =
(a − bx)2
.
R. f ′ (x) = esin x cos x .
R. f ′ (x) = ex (cos x + sin x) .
R. f ′ (x) = natan nx ln(a) sec2 (nx) .
R. f ′ (x) = x1/x (1 − ln x)/x2 .
√
′
R. f (x) =
x
x
(2 + ln x)
√
.
2 x
R. f ′ (x) = sinx x(x cot x + ln(sin x)) .
78. f (x) = sintan x x
R. f ′ (x) = sintan x x(1 + x sec2 x ln(sin x)) .
79. f (x) = sintan x x
q
sin x+1
80. f (x) = sin
x−1
R. f ′ (x) = sintan x x(1 + x sec2 x ln(sin x)) .
R. f ′ (x) = −
cos x
q
.
sin x+1
(sin x − 1)2 sin
x−1
1.1. DERIVADAS DIRECTAS
√
81. f (x) = arcsin sin x
cos x
.
R. f ′ (x) = p
2 sin x − sin2 x
82. f (x) = earctan x
R. f ′ (x) =
2
.
ex + e−x
p
√
x + 1 + x2
′
√
.
R. f (x) =
2 1 + x2
83. f (x) = arctan ((ex − e−x )/2)
84. f (x) =
85. f (x) =
p
s
x+
86. f (x) = sinm x cosn x
87. f (x) = cot2 (sin x)
88. f (x) =
R. f ′ (x) =
√
1 + x2
√
1− x
√
1+ x
p √
x x
R. f ′ (x) = −
91. f (x) =
√
x2 + 1 − x
)
92. f (x) = ln( √
x2 + 1 + x
93. f (x) = xn (a + bx)m
p
√
94. f (x) = sin( x) + sin(x)
1
p
.
x)( x(1 − x))
R. f ′ (x) = − cos x cot(sin x) csc2 (sin x) .
√
3 x3/2
R. f ′ (x) =
.
4x
1
−2 − p
1/x
.
R. f ′ (x) = qp
4
1/x + 1/xx2
1
1+ √
2 x
1+ p
√
2 x+ x
R. f ′ (x) = q
p
√ .
2 x+ x+ x
q
p
√
x+ x+ x
q
p
√
2+ 2+ x
2(1 +
√
R. f ′ (x) = sinm−1 x cosn−1 x(m cos2 x − n sin2 x) .
q
p
89. f (x) = 1/x + 1/x
90. f (x) =
earctan x
.
1 + x2
1
R. f ′ (x) = q
p
√ p
√ √ .
8 2+ 2+ x 2+ x x
−2
R. f ′ (x) = √
.
x2 + 1
R. f ′ (x) = (xn (a + bx)m )(
mb
n
+
).
x a + bx
√
cos(x)
cos( x)
+p
R. f (x) = 1/2( √
).
x
sin(x)
′
95. f (x) = sin(nx) cos(mx)
R. f ′ (x) = n cos(mx) cos(nx) − m sin(mx) sin(nx) .
96. f (x) = tan(nx) cot(mx)
R. f ′ (x) = n cot(mx) sec2 (nx) − m csc2 (mx) tan(nx) .
97. f (x) = enx ln(mx)
R. f ′ (x) =
98. f (x) = 2x 3x 5x
√
99. f (x) = 3
x
√
5
x
enx (1 + nx ln(mx))
.
x
R. f ′ (x) = 30x ln(30) .
√
′
R. f (x) = −
(3/5)
x
ln(5/3)
√
.
2 x
1.2. DERIVADAS IMPLÍCITAS
100. f (x) =
s
sin(e2x )
√
ln( x)
1.2. Derivadas implı́citas
R. f ′ (x) =
2e2x cos(e2x ) ln(x) − sin(e2x )
r
.
√
sin(e2x )
2
2x ln x
ln x