This document provides 98 examples of functions and their derivatives. The functions include polynomials, trigonometric functions like sine, cosine, tangent, inverse trigonometric functions, exponential functions, logarithmic functions, and combinations of these functions.

1. ejerciciosyexamenes.com
DERIVADAS
1. y = 5x6-3x5+3x3-2 2. y = x-4+2x-3+x-4
3
3. y = 3 x10 + 2 x + 4. y = 3 . x 3 - π . x + 3
x
2
5. y = 4 senx - 3 cosx 6. y = 2 x + + x5
x
π
7. y = 4x3 + 2x3 - x3 + 4 8. y = . cos x - 3 x
2
9. y = cos(3x) 10. y = cos2(x3)
11. y = sen (3x2-2x) 12. y = cos(x2)
13. y = sen3(2x2) 14. y = cos4(3x4)
15. y = 3 sen2(2x-3) 16. y = cos5(3x2)
17. y = cos (senx) 18. y = cos2(sen(3x))
19. y = 3 cos2 x 20. y = 3 cos 2 ( x2 )
21. y = x2 - 3x 22. y = 3 ( x 2 - 3x )2
23. y = 2 ( x - 3x )
3
24. y = 3 sen 2 x
25. y = 5 sen(3x) 26. y = 3x - senx
(
27. y = 3 x 2 - 1 - x2 ) 3
28. y = sen ( 3 x2 - 5x )
29. y = sen3 x + (x - 1 )3 30. y = cos3 x2 - 3 x ( )
x 5
31. y = 32. y =
5 x
4 3
x - 3x x -3
33. y = 34. y =
4 x
( x - 3x )2
4
(x - 1 )3
35. y = 36. y =
3 3x
2
x 3x
37. y = 2 38. y =
x -1 x
3 x
39. y = 40. y =
x 3x
41. y = 3 3 x 2 - senx 42. y = l n (3x - 1)
43. y = l n ( x - 3x) 2
44. y = l n x - 2
45. y = log 2 (3 x 2 ) 46. y = e x
2
47. y = 2 x 48. y = e x - 2x
2
49. y = 3sen x 50. y = tg ( x3 )
51. y = 3 e x - 3x 52. y = ecosx
2
53. y = 3 tg 2 x 54. y = ( x2 - 1) .(x - 1)
55. y = x2 . l n x 56. y = e x . cos x
2
57. y = x4 . e 3 x 58. y = ex
4 -3 2
x
. sen x

2. ejerciciosyexamenes.com
1
59. y = l n x2 . esen x 60. y =
ln x
3
 2-3 
61. y =  x2
  62. y = l n x . e x - senx
2

 x +1 
2
 ln 2 
63. y = ( cos 3x - sen x
2 3
).e x3
64. y =  3 x 
 
 x -2 
x
lnx +l n x
65. y = 66. y = e 2
3x x - senx
 sen x  sen x
67. y = l n   68. y =
 e 
x
x- 1
70. y = ( sen ( e3x ) )
4 2
3 - 2 x + 3x - 2
69. y = x
2
. cos x
2 x -1
71. y = arctg ( x 2 ) 72. y = arcsen x 3
73. y = l n ( sec x) 74. y = arctg (l n x)
75. y = arcsen x . ecosx 76. y = arctg ( e 3x )
 tgx   x +1 
77. y = l n  3 x2  78. y = arcsen  
e   e 
x
79. y = l n ( arctg(5x) ) 80. y = arctg x 3
81. y = 5 arctg2 (senx) 82. y = 3arctg( x )
2
arcsen (3x - 2) sen x - tg x
83. y = 84. y =
x2 4x - 3
2
85. y = x senx 86. y = (sen x )x
3
87. y = ( cos x )x - x 88. y = 4arctg ( l n x)
x2
 x+ 2 
3
 3x

89. y = l n   90. y =  e
 senx  
 x 
2
 
 senx-e x   tg x 
91. y = 10  3x- cosx 
  92. y = cos  
 
 sen (l n x) 
 
x
cosx
 tgx 
93. y = 94. y = cos2 (4 e x ) . l n  x 2 
(l n x )3  3 
2 x l n ( cos x)
sen ( e )
95. y = 3 96. y = e
arctg (cosx) 5 cos x
98. y = ( tg ( e x ) + x 2 )
cos( e x ) . x x
97. y = tgx
e
arctg e x . cos x l n cos x
99. y = 100. y =
l n ( x 2 - x) sen ( ecosx )

3. ejerciciosyexamenes.com
SOLUCIONES Todas las soluciones se dan sin simplificar
1. y′ = 30 x 5 - 15 x4 + 9 x 2 2. y′ = - 4 x-5 - 6 x -4 + 1
1 3
3. y′ = 30 x9 + - 2 4. y′ = 3 3 . x2 - π
x x
1 2
5. y′ = 4 cos x + 3 sen x 6. y′ = - 2 + 5 x4
x x
π 3
7. y′ = 15 x 2 8. y′ = - sen x -
2 2 x
9. y′ = - 3 sen(3x) 10. y′ = - 2 cos( x ) . sen( x3 ) . 3 x 2
3
11. y′ = cos (3 x2 - 2x) . (6x - 2) 12. y′ = - sen( x 2 ) . 2x
13. y′ = 3 sen 2 (2 x 2 ) . cos(2 x 2 ) . 4x 14. y′ = 4 cos 3 (3 x4 ) . ( - sen(3 x4 ) ) . 12 x 3
15. y′ = 6 sen(2x - 3) . cos(2x - 3) . 2 16. y′ = 5 cos 4 (3 x 2 ) . (-sen(3 x 2 )) . 6x
17. y′ = - sen(senx) . cosx 18. y′ = 2 cos(sen3x) . (-sen(sen3 x)) . cos 3x . 3
- 2 cosx . senx - 2 cos( x 2 ) . sen( x 2 ) . 2x
19. y′ = 20. y′ =
3 3 cos 4 x 3 3 cos 4 ( x2 )
2x - 3 2 ( x 2 - 3x) (2x - 3)
21. y′ = 22. y′ =
2 x2 - 3x 3 3 ( x 2 - 3x )4
(
23. y′ = 3 2 x - 3x . 
2
)
 1
 x

- 3  24. y′ =

2 senx cosx
3 3 sen 4 x
3 cos 3x 3 - cosx
25. y′ = 26. y′ =
5 5 sen4 (3x) 2 3x - senx
27. y′ = 3 ( 3x 2
-

.  6x -
1 - x2
 )
2- 2x 
 28. y′ = cos

( )
3 x2 - 5x .
6x - 5
3 x 2 - 5x
 2 1 - x2  2
3 2
x cosx + 3 (x - 1 )2
29. y′ = sen 30.
2 sen 3 x + (x - 1 )3
( ) (
y′ = - 3 cos 2 x 2 - 3 x . sen x 2 - 3 x .  2x -) 3 

 2 x 
1 5
31. y′ = 32. y = - 2
5 x
3
4 -3 3
33. y′ = x 34. y′ = 2x + 2
4 x
2
4
2 ( - 3x ) . (4 x - 3)3
3 (x - 1 )2 . 3x - 3 (x - 1 )3
35. y′ = x 36. y′ =
3 (3 x )2
3
2 2 . x - 3x
2x ( x - 1) - x . 2x 2 3x
37. y′ = 38. y′ =
( x2 - 1 )2 x2

4. ejerciciosyexamenes.com
3
3x - x
1 -3 2 3x
39. y′ = 2
. 40. y′ =
3 x 3x
2
x
6 x - cosx 3
41. y′ = 42. y′ =
2
3 3 2
(3 x - senx ) 3x - 1
1
2x - 3
43. y′ = 44. y′ = 2 x - 2
x 2 - 3x x-2
6x
45. y′ = 2 46. y′ = e x . 2x
2
3 x . l n2
47. y′ = 2x . l n2 48. y′ = e x - 2x . (2x - 2)
2
3 x2
49. y′ = 3sen x . cos x . l n3 50. y′ =
cos2 ( x3 )
- cosx senx
51. y′ = 3 e x -3x . (2x - 3) 52. y′ = e cosx
2
2 e
6 tgx
53. y′ = 54. y = 2x (x - 1) + ( x 2 - 1)
cos 2 x
2
55. y′ = 2x . l n x + x 56. y′ = e x 2x cos x - e x senx
2 2
x
57. y′ = 4 x . e + x4 e3x . 3 58. y′ = e x - 3 x . (4 x 3 - 6x) sen x + e x -3 x cos x
3 3x 4 2 4 2
1
- 2 x
2x x
59. y′ = . esen x + l nx 2 e sen x cos x 60. y′ =
x 2
(l n x )2
2
 2 - 3  2x ( x 2 + 1) - ( x2 - 3) 2x
61. y′ = 3  x2
  .

 x +1  ( x2 + 1 )2
1
62. y′ = . ex -senx + l nx . e x - senx (2x - cosx)
2 2
x
63. y′ = ( 2 cos 3x (-sen3x) 3 - 3 sen 2 x cosx ) . e x + ( cos2 3x - sen 3 x) e x 3 x 2
3 3
2x 3
( - 2) - l nx2 3 x2
 l n x  x2 x
2
64. y′ = 2  3
  .

 x -2  ( x3 - 2 )2
1 x  x 1  2
x
3 - ln x 3 l n3  e +  ( x - senx) - ( ex + l nx) (2x - cosx)
66. y′ = 
x 
65. y′ = x x 2
(3 ) ( x 2 - senx )2
x
cos x e x - e x sen x 1 cosx (x - 1) - senx
67. y′ = e 2
68. y′ = .
senx ( ex ) sen x (x - 1 )2
2
x -1

5. ejerciciosyexamenes.com
(12 x 3 - 4x + 3) (2x - 1) - (3 x4 - 2 x2 + 3x - 2) . 2
69. y′ =
(2 x - 1 )2
70. y′ = 2 ( sen ( e 3x ) ) . cos ( e
3x
) . e3x 3 cosx - (sen( e3x ) ) . senx
2
2x 3 x2
71. y′ = 72. y′ =
1 + x4 1 - x6
senx 1
2
73. y′ = cos x 74. y′ = x
sec x 1 + (l n x )2
cos x 3x
.3
75. y′ = e 2 - arcsen x ecos x sen x 76. y′ = e 3x 2
1- x 1 + (e )
2 x x
3
e x - tgx 3 x 2 6x e - (x + 1) e
2 e
e3 x . cos2 x ( e x )2
77. y′ = 78. y′ =
tgx ( )
e 3x
2 2
1- 
 x+ 1 

2
 e 
x
3 x2
1 5 2 x3
79. y′ = 80. y′ =
arctg(5x) 1 + (5x )2 1 + x3
cos x 2x
81. y′ = 10 arctg (senx) 82. y′ = 3arctg ( x )
2
2
ln3
1 + sen x 1 + x4
3 x2
- arcsen (3x - 2) 2x
1 - (3x - 2 )2
83. y′ =
x4
 1  4
 cosx -  4x - 3 - (sen x - tg x)
y′ =  cos x 
2
2 4x - 3
84.
4x - 3
y′ =  cosx . l n x +
senx 
85.   . xsenx
 x 
 2
cosx 
y′ =  2x l n (senx) + x  (sen x ) x
2
86.  
 senx 
 ( 3 - x) senx 
87. y =  (3 x 2 - 1) l n cos x - x  . (cosx )x - x
3
 
 cosx 
1
88. y′ = 4 arctg( l n x) . x . ln4
1 + (l n x )2
 x + 2  x 2 - 2x (x + 2)
2
1
89. y′ = 3
.3   .
 x+2   x 
2
x4
 
 x 
2

6. ejerciciosyexamenes.com
2
  e3x 3x
2 senx 3 e senx - e cosx
3x
  e3x  x
90. y′ =  2x l n x
 
+ x
 senx  .
  senx  
  e3x sen 2 x   
 senx-e x  x x
(cosx - e ) (3x - cosx) - (senx - e ) (3 + senx)
91. y′ = 10 3x- cosx  . l n 10




(3x - cosx )2
1 1 1
. . sen(l n x) - tg x . cos(l n x) .
 tg x  cos x 2 x 2
x
92. y′ = - sen   .
 sen (l n x)  2
  sen (l n x)
 cosx  cosx 2 1
 - senx l n nx +  x (l n x ) - x cosx 3 (l n x )
3
93. y′ = 
x  x
6
(l n x )
94.
2
3 x - tgx . x2 . 2x . l n 3
3
 tgx 
2
x 2
y′ = 2 cos (4 e x ) . (-sen(4 e x ) . 4 e x . l n  x2  + cos 2 (4 e x ) . 3 cos x
 3  tgx
2
3x 2 ( )
l n (cosx) - senx
e . 5 cosx - el n (cosx) . 5cosx (-senx) l n 5
1 cosx
95. y = .
2 e
l n (cosx)
( 5 cosx )2
5cosx
96.
1
2 sen( e x ) . cos( e x ) . e x . arctg(cosx ) - sen2 ( e x ) . . (-senx)
1 1 + cos 2 x
y′ = .
sen2 ( e x ) arctg2 (cosx)
3 3
arctg ( cos x)
- sen( e x ) ex x + cos( ex ) tgx 1 1
. e - cos( e x ) . x . . etgx .
cos( e x ) . x 2 e tgx
cos 2 x
97. y′ = tgx
e
 e  x
 + 2x 
98. y′ =  ln ( tge + x 2 ) + x cos e 2  . ( tg ( x ) + 2 )x
2 x
x ( )
 x  e x
tge + x
 
 
99.
 e
x
 2x - 1

 1 + ( e x )2 cosx - arctg e senx  l n ( x - x) - (arctg e cosx) x 2 - x
x 2 x

. 
1 
y′ = x
arctg e . cos x l n ( x 2 - x)
2
2
l n ( x2 - x)

7. ejerciciosyexamenes.com
1 - senx
sen( ecosx ) - l n cos x cos( ecosx ) e cosx (-senx)
100. y′ = cosx 2 cosx
sen 2 ( ecosx )