DERIVADAS DE FUNCIONES 1.
f ( x) = x 5 − 4 x 3 + 2 x − 3
f ' ( x ) = 5 x 4 − 12 x 2 + 2
2.
f ( x) = x 3 + 5 x 2 − 7 x + 1
f ' ( x ) = 3 x 2 + 10 x − 7
3.
f ( x) = 5 x 4 − 7 x 3 + 3x 2 − x + 4
f ' ( x ) = 20 x 3 − 21 x 2 + 6 x − 1
4.
f ( x) =
5.
f ( x) = ax 3 + bx 2 − cx + d
f ' ( x ) = 3ax 2 + 2bx − c
6.
f ( x) = ( x 3 + 5 x 2 ) ⋅ (7 x − 1)
f ' ( x ) = 28 x 3 + 102 x 2 − 10 x
7.
f ( x) = x ⋅ (2 x 5 − 3) ⋅ (3 x + 2)
f ' ( x ) = 42 x 6 + 24 x 5 − 18 x − 6
8.
f ( x) =
9.
x2 f ( x) = x+2
10. f ( x) =
2 3 5 2 7 1 x + x − x+ 3 2 9 5
x −1 x +1
f '( x) = 2 x 2 + 5 x −
f '( x) =
7 9
2 ( x + 1) 2
x 2 + 4x f '( x) = ( x + 2) 2
x2 − x x2 +1
f '( x) =
x 2 + 2x − 1 ( x 2 + 1) 2
11. f ( x) = (3 x − 2) 2
f ' ( x ) = 6 ⋅ ( 3 x − 2)
3 12. f ( x) = ( x 2 − 2 x + 1) 5 2
3 f ' ( x ) = 5 ⋅ ( x 2 − 2 x + 1) 4 ⋅ ( 3 x − 2) 2
13. f ( x) = (3 x 2 + 2 x − 5) 3
f ' ( x ) = 3 ⋅ ( 3 x 2 + 2 x − 5) 2 ⋅ ( 6 x + 2)
14. f ( x) = (3 x − 2) 2 ⋅ ( x + 7) 5
f ' ( x ) = ( 3 x − 2) ⋅ ( x + 7 ) 4 ⋅ ( 21 x + 32)
2 15. f ( x) = ( x 2 − 1) 2 ⋅ ( x 3 + 1) 5 3
x −1 16. f ( x) = x + 1
2 19 f ' ( x ) = 2 x ⋅ ( x 2 − 1) ⋅ ( x 3 + 1) 4 ⋅ x 3 − 5 x + 2 3 3
3
x2 − x 17. f ( x) = 2 x +1
f '( x) = 4
6 ⋅ ( x − 1) 2 ( x + 1) 4 3
x2 − x x2 + 2x − 1 ⋅ f ' ( x ) = 4 ⋅ 2 ( x 2 + 1) 2 x +1
18. f ( x) =
( x − 1) 3 ( x + 1) 2
f '( x) =
( x − 1) 2 ⋅ ( x + 5) ( x + 1) 3
19. f ( x) =
( x + 2) 3 (1 − 2 x) 2
f '( x) =
( x + 2) 2 ⋅ (11 − 2 x ) (1 − 2 x ) 3
20. f ( x) =
( x − 1) 3 ⋅ ( x − 2) x−3
f '( x) =
( x − 1) 2 ⋅ ( 3 x 2 − 16 x + 19) ( x − 3) 2
1
21. f ( x) = L x
f '( x) =
1 x
22. f ( x) = L (3 x)
f '( x) =
1 x
23. f ( x) = L ( x 2 )
f '( x) =
2 x
24. f ( x) = L ( x + 1)
f '( x) =
1 x +1
25. f ( x) = L (3 x + 2)
f '( x) =
3 3x + 2
26. f ( x) = L ( x 2 − 1)
f '( x) =
2x x2 − 1
27. f ( x) = log 5 x
f '( x) =
1 ⋅ log 5 e x
28. f ( x) = log 3 ( x 3 − 2 x)
f '( x) =
3x2 − 2 ⋅ log 3 e x3 − 2x
f '( x) =
42 x 6 + 24 x 5 − 18 x − 6 x ⋅ ( 2 x 5 − 3) ⋅ ( 3 x + 2 )
f '( x) =
21 x + 32 ( 3 x − 2) ⋅ ( x + 7 )
f '( x) =
2 x −1
f '( x) =
8 x −1
f '( x) =
11 − 2 x ( x + 2) ⋅ (1 − 2 x )
34. f ( x) = L( L x)
f '( x) =
1 x⋅Lx
35. f ( x) = L( L( L x))
f '( x) =
1 x ⋅ L( L x ) ⋅ L x
36. f ( x) = L3 x
f '( x) =
3 ⋅ L2 x x
37. f ( x) = L2 ( x 2 + 1)
f ' ( x) =
4 x ⋅ L ( x 2 + 1) x2 + 1
(
29. f ( x) = L x ⋅ (2 x 5 − 3) ⋅ (3 x + 2)
[
30. f ( x) = Ln (3 x − 2) 2 ⋅ ( x + 7) 5
31. f ( x) = Ln
x −1 x +1
x − 1 4 32. f ( x) = Ln x + 1 33. f ( x) = L
( x + 2) 3 (1 − 2 x) 2
]
)
38. f ( x) = x −2
2
2
f ' ( x ) = − 2 ⋅ x −3
2
1 x3
f ( x) = −
40. f ( x) = (3 x 2 − 3 x + 4) −4
f '( x) = −
12( 2 x − 1) ( 3 x 2 − 3 x + 4) 5
f '( x) = −
5 ⋅ ( 2 x + 1) ( x 2 + x + 1) 6
39. f ( x) =
41. f ( x) =
1 ( x + x + 1) 5 2
42. f ( x) = x
f '( x) = 3
43. f ( x) = x + 3x
f '( x) =
44. f ( x) = 3 ( x 2 + 1) 2
f '( x) =
45. f ( x) = 5 2 ⋅ L x + 3
f '( x) =
46. f ( x) =
x −1 x +1
47. f ( x) = L
2 x
3 ⋅ ( x 2 + 1) 2 ⋅ x3 + 3x 4x 3⋅ x2 + 1 3
2 4
5 x ⋅ 5 (2 ⋅ L x + 3)
(
9 12 6 1 + − + 6 5 4 ( x + 2) ( x + 2) ( x + 2) ( x + 2) 3
f ' ( x) =
48. f ( x) = L x + 1 + x 2
50. f ( x) =
1
9 3 2 1 − + − 5 4 3 5( x + 2) ( x + 2) ( x + 2) 2( x + 2) 2 f '( x) = −
49. f ( x) =
3 x4
)
1 x −1
f '( x) =
1 L2 x
f '( x) = −
2
1 1 + x2 2 x ⋅ L3 x
( x − 1) 5 ⋅ 7 ( x + 1) 3 ⋅ ( x + 2) 5 ( x + 3)
7 2
5 3 5 5 ( x − 1) ⋅ 7 ( x + 1) ⋅ ( x + 2) 3 5 7 f '( x) = + + − ⋅ 7 x − 1 7( x + 1) 7( x + 2) 2( x + 3) ( x + 3) 2
(
51. f ( x) = x − 2 x + 2 ⋅ L 1 + x
52. f ( x) = L
)
f '( x) =
a2 + x2 + x
f '( x) =
a2 + x2 − x
53. f ( x) = 3 x
x x ⋅ (1 + x ) 2 a2 + x2
f ' ( x) = 3 x ⋅ L 3
3
54. f ( x) = 3 x ⋅ 5 x
f ' ( x ) = 15 x ⋅ Ln 15
55. f ( x) = e x +3
f '( x ) = e x+3
56. f ( x) = e x
2
57. f ( x) = e
2 x +5
−1
f ' ( x) = 2 x ⋅ e x
59. f ( x) =
f '( x) = −
2
60. f ( x) = 5e − x
2
2 x+5
f ' ( x ) = 2 x ⋅ 10 2 x ⋅ (1 + x ⋅ L 10)
1 5x
−1
1 ⋅e 2x + 5
f '( x) =
58. f ( x) = x 2 ⋅ 10 2 x
2
+3
2 x ⋅ Ln 5 5x
2
f ' ( x ) = −10 x ⋅ e − x
2
+3
e x + 3 ⋅ ( x − 2) x3
61. f ( x) =
e x+ 3 x2
f ' ( x) =
62. f ( x) =
x3 ex
f ' ( x ) = x 2 ⋅ e − x ⋅ (3 − x )
63. f ( x) = ( x 2 − 2 x + 2) ⋅ e − x
f ' ( x ) = − e − x ⋅ ( x 2 − 4 x + 4)
64. f ( x) = 2e − 2 + 1 + L x
5 ⋅ L4 x f '( x) = + x 3 ⋅ 3 ( 2e x − 2 x + 1) 2
65. f ( x) = x x
f ' ( x ) = ( L( x ) + 1) ⋅ x x
x
3
66. f ( x) = x x
x
2
f '( x) = −
x
f '( x) =
69. f ( x) = x L x
1 70. f ( x) = 1 + x 71. f ( x) = x x
2e x − 2 x ⋅ L 2
f ' ( x ) = (2 ⋅ L( x ) + 1) ⋅ x x
67. f ( x) = x x 68. f ( x) = x
5
(L( x ) − 1) ⋅ x x 1
2⋅ x
2
+1
x
2
⋅ (L( x ) + 2 ) ⋅ x
x
f ' ( x ) = 2 ⋅ L( x ) ⋅ x Lx −1 x
x
1 1 1 f ' ( x ) = L 1 + − ⋅ 1 + x x + 1 x
x
f ' ( x ) = [( L( x ) + 1) ⋅ x ⋅ L( x ) + 1] ⋅ x x −1 ⋅ x x
72. y = 3 ⋅ sen x
y' = 3 ⋅ cos x
73. y = 2 ⋅ sen ( x 3 + 5)
y' = 6 x 2 ⋅ cos ( x 3 + 5)
74. y = 3 ⋅ sen (L x)
y' =
4
3 ⋅ cos (L x ) x
x
3(1 + e x ) ⋅ cos x + e x
75. y = 3 ⋅ sen x + e x
y' =
76. y = sen 2 x
y' = 2 ⋅ sen x ⋅ cos x = sen 2 x
77. y = a ⋅ cos x
y' = − a ⋅ sen x
78. y = a ⋅ cos 2 x
y' = − a ⋅ sen 2 x
79. y = a ⋅ cos 2 x 2
y' = −2ax ⋅ sen 2 x 2
80. y = a ⋅ cos(sen x)
y' = − a ⋅ sen (sen x ) ⋅ cos x
81. y = a ⋅ 1 − cos x
y' =
82. y = 3 ⋅ sen x + 5 ⋅ cos x
y' = 3 ⋅ cos x − 5 ⋅ sen x
83. f ( x) = tg (2 x + 3)
f ' ( x ) = 2 ⋅ sec 2 ( 2 x + 3)
84. f ( x) = tg (e − x + x)
f '( x) =
85. f ( x) = tg x − cotg x
f ' ( x ) = sec 2 x + cosec 2 x
86. f ( x) = tg (cos 2 x)
f ' ( x ) = −2 ⋅ sec 2 (cos 2 x ) ⋅ sen 2 x
87. f ( x) = ctg (sen x)
f ' ( x ) = −cosec 2 (sen x ) ⋅ cos x
88. f ( x) =
1 − cos x 1 + cos x
89. f ( x) =
sen x + cos x sen x − cos x
2⋅ x + ex
a ⋅ sen x 2 ⋅ 1 − cos x
f '( x) =
− e−x + 1 cos 2 (e − x + x )
1 x ⋅ sec 2 2 2
f '( x) = −
2 (sen x − cos x ) 2
90. f ( x) = e x ⋅ cos x
f ' ( x ) = e x ⋅ (cos x − sen x )
91. f ( x) = (3 − 2 ⋅ sen x) 5
f ' ( x ) = −10 cos x ⋅ ( 3 − 2 ⋅ sen x ) 4
92. f ( x) = ctg x − ctg a
f '( x) = −
93. y = 2 x + 5 ⋅ cos 5 x
y' = 2 − 25 ⋅ cos 4 x ⋅ sen x
94. y = −
1 6 ⋅ (1 − 3 cos x) 2
1 1 − 95. y = 3 3 cos x cos x 96. y =
3 ⋅ sen x − 2 ⋅ cos x 5
y' =
1 2 ⋅ sen x ⋅ ctg x 2
sen x (1 − 3 cos x ) 3
sen 3 x y' = cos 4 x
y' =
5
3 ⋅ cos x + 2 ⋅ sen x 2 ⋅ 5 ⋅ ( 3 ⋅ sen x − 2 ⋅ cos x )
97. y = sec 2 x + cosec 2 x 98. f ( x) = a ⋅ ctg
y' = −
x a
16 ⋅ cos 2 x sen 3 2 x
f ' ( x ) = −cosec 2
1 1 99. y = tg x − ⋅ tg 3 x + ⋅ tg 5 x 3 5
y' = 1 + tg 6 x
100.
y = arcsen 2 x
y' =
101.
y = arcsen x
y' =
102.
y = arcsen x 2
y' =
103.
y = 1 + arcsen x
y' =
104.
y = e arcsen x
y' =
x a
2 1 − 4x 2 1 2⋅ x − x2 2x 1 − x4 1 1 ⋅ 2 ⋅ 1 + arcsen x 1 − x 2
1 1− x
2
⋅ e arcsen x
x y' = e arcsen x ⋅ 1 + 1 − x2
105.
y = x ⋅ e arcsen x
106.
y = arcsen
1 x
y' = −
107.
y = arccos e x
y' = −
108.
y = arctg x
y' =
1 2 ⋅ x ⋅ (1 + x )
109.
y = arcctg
y' =
1 1 + x2
110.
y=
1− x 1+ x
arccos x
y' =
1− x2
1 x ⋅ x2 − 1 ex 1 − e 2x
x ⋅ arccos x − 1 − x 2 2
(1 − x ) x a
111.
f ( x) = a 2 − x 2 + a ⋅ arcsen
112.
f ( x) = L(e x + 5 ⋅ sen x − 4 ⋅ arcsen x)
113.
f ( x) = arctg ( L x) + L (arctg x)
f '( x) =
f '( x) =
3 2
a−x a+ x (e x + 5 ⋅ cos x ) ⋅ 1 − x 2 − 4
(e x + 5 ⋅ sen x − 4 ⋅ arcsen x ) ⋅ 1 − x 2
f '( x) =
6
1 1 + 2 2 x ⋅ 1 + ( L x) (1 + x ) ⋅ arctg x
(
)
−x
114.
f ( x) = arcsen (1 − x) + 2 x − x 2
115.
f ( x) = arctg
116.
tg x f ( x) = − 2 ⋅ arctg −x 2
4 + 3 ⋅ tg 2 x f '( x) = − 2 + tg 2 x
117.
y = 2 arcsen 3 x + (1 − arccos 3 x) 2
y'=
118.
2 x 1 x −1 y= ⋅ arctg + ⋅L 3 2 6 x +1
x2 y' = 2 ( x + 2) ⋅ ( x 2 − 1)
119.
y = Ln
120.
f ( x) = sen (cos( tg (arctg (arccos(arcsen x)))))
121.
f ( x) = x 4 + 3 x 2 − 6
f '( x) = 4 x 3 + 6 x
122.
f ( x) = 6 x 3 − x 2
f ' ( x ) = 18 x 2 − 2 x
123.
f ( x) =
x5 x2 − −x a +b a −b
f '( x) =
5x4 2x − −1 a+b a−b
124.
f ( x) =
x3 − x2 +1 5
f '( x) =
3x2 − 2x 5
125.
f ( x) = 2ax 3 −
f '( x) =
x ⋅ sen a 1 − x ⋅ sen a
f '( x) =
1 + sen x + 2 ⋅ arctg sen x 1 − sen x
7 2
y' =
x2 +c b
f ( x) = 6 x + 4 x + 2 x
127.
f ( x) = 3x + 3 x +
128.
f ( x) =
3 ⋅ ( 2arcsen 3 x ⋅ Ln 2 + 2 ⋅ (1 − arccos 3 x) )
1 − (3 x)2
2 cos x ⋅ sen x
f '( x) = 1
2x b
5 2
3 2
f ' ( x ) = 21 x + 10 x + 2
1 x
f '( x) =
( x + 1) 3 x
sen a 1 − 2 x ⋅ sen a + 2 ⋅ x 2 ⋅ sen 2 a
f ' ( x ) = 6ax 2 −
5 2
126.
2x − x2
f '( x) =
3 2
3 1 1 + − 2 2 3 x 2 x 3 x 3( x + 1) 2 ⋅ ( x − 1) 2⋅ x
5 2
129.
f ( x) =
x a x2 b2 + + + a x b2 x2
f '( x) =
1 a 2 x 2b 2 − + − 3 a x 2 b2 x
130.
f ( x) = 3 x 2 − 2 x + 5
f '( x) =
2 1 1 ⋅3 − 3 x x
131.
f ( x) =
f '( x) =
− 5 3 1 − ⋅ ax 3 − ⋅ bx 2 + ⋅ x 6 3 2 6
2
3 a ⋅ x2 b x + − 3 x x x x
7
5
7
132.
f ( x) = (1 + 4 x 3 )(1 + 2 x 2 )
f ' ( x ) = 4 x ⋅ (10 x 3 + 3 x + 1)
133.
f ( x) = x(2 x − 1)(3 x + 2)
f ' ( x ) = 2 ⋅ (9 x 2 + x − 1)
134.
f ( x) =
2x 4 b2 − x2
f '( x) =
4 x 3 ( 2b 2 − x 2 ) (b 2 − x 2 ) 2
135.
f ( x) =
a−x a+x
f ' ( x) =
− 2a (a + x ) 2
136.
f (t ) =
t3 1+ t 2
f ' (t ) =
t 2 ⋅ (3 + t 2 ) (1 + t 2 ) 2
137.
f ( s) =
( s + 4) 2 s+3
f ' ( s) =
( s + 4) ⋅ ( s + 2) ( s + 3) 2
138.
f ( x) =
xp xm − am
f ' ( x) =
x p −1 ⋅ ( p − m ) x m − pa m ( x m − a m )2
139.
f ( x) = (2 x 2 − 3) 2
f ' ( x ) = 8 x ⋅ ( 2 x 2 − 3)
140.
f ( x) = ( x 2 + a 2 ) 5
f ' ( x ) = 10 x ⋅ ( x 2 + a 2 ) 4
141.
f ( x) = x 2 + a 2
f '( x) =
142.
f ( x) = ( x + a ) x − a
f '( x) =
143.
f ( x) =
1+ x 1− x
f '( x) =
144.
f ( x) =
2x2 −1
f ' ( x) =
x 1+ x2 3
f ( x) = (1 + x )
146.
f ( x) = x + x + x
147.
f ( x) =
148.
f ( x) = a ⋅ cos 2 x
149.
r (Φ ) = a ⋅ sen 3
f '( x) =
)
x x2 + a2
3x − a 2 x−a 1 (1 − x ) ⋅ 1 − x 2
4x 2 + 1 x 2 ( 1 + x 2 )3
1 f '( x) = 1 + 3 x
3
145.
(
2
1 1 ⋅ 1 + ⋅ 1 + 2 x 2 x+ x 2 x+ x+ x 1
sen x 1 + cos x
Φ 3
f ' ( x) =
1 1 + cos x
f '( x) =
− a ⋅ sen 2 x cos 2 x
r '(Φ ) = a ⋅ sen 2
8
Φ Φ ⋅ cos 3 3
2
150.
x f ( x) = a ⋅ 1 − cos 2 2
151.
f ( x) =
tg x − 1 sec x
f ' ( x ) = cos x + sen x
152.
f ( x) =
1 ⋅ tg 2 x 2
f ' ( x ) = tg x ⋅ (1 + tg 2 x )
153.
f ( x) =
1 3 ⋅ tg x − tg x + x 3
f ' ( x ) = tg 4 x
154.
f ( x) = tg(Ln x)
f '( x) =
155.
f ( x) = sen (cos x)
f ' ( x ) = −sen x ⋅ cos(cos x )
156.
x x tg + ctg 2 f ( x) = 2 x
x x 2 x cos x + sen 2 x ⋅ tg + ctg 2 2 f ' ( x) = − 2 2 x ⋅ sen x
157.
f ( x) = Ln
f ' ( x ) = 2a ⋅ sen 3
x2 + 1 − x
x x ⋅ cos 2 2
sec2 (Ln x) x
f ' ( x) = −
2
x +1 + x
2 2
x +1
a + a2 + x2 x
f '( x) =
a2 + x2 x
f ' ( x) =
x2 + a2 x2
158.
f ( x) = a 2 + x 2 − a ⋅ Ln
159.
f ( x) = Ln( x + x 2 + a 2 ) −
160.
f ( x) = −
161.
f ( x) =
sen x 2 cos 2 x
f '( x) =
162.
f ( x) =
1 2 ⋅ tg x + Ln ( cos x ) 2
f ' ( x ) = tg 3 x
163.
f ( x) = e 4 x+5
164.
f ( x) = 7 x
2
+2 x
f '( x) = 7 x
165.
f ( x) = c a
2
− x2
f '( x) = −2 x ⋅ c a
x2 + a2 x
cos x 1 x + ⋅ Ln tg 2 2sen x 2 2
1 2 ⋅ sen 3 x
f '( x) =
1 + sen 2 x 2 cos 3 x
f ' ( x) = 4 ⋅ e 4 x+5
9
2
+2 x
⋅ (2 x + 2) ⋅ Ln 7 2
− x2
⋅ Ln c