Derivadas

100 DERIVADAS RESUELTAS

Hallar las derivadas simplificadas de las siguientes funciones:
22. y=(x2+1)100

1.

y=3

(y’=0)

2.

y=x

(y’=1)

3.

y=5x

(y’=5)

4.

y=x

(y’=1)

5.

y=x4+x3+x2+x+1

(y’=4x3+3x2+2x+1)

4

3

2

23. y =

3

6.

y= 4x -x +3x -7

7.

2

1
1
1
y = − x5 + 4x4 − x3 + x 2 − 3
5
6
2

(y’=16x -3x +6x)

12
⎛
⎞
43
⎜ y' = − x + 16x − x + x ⎟
2
⎝
⎠

8.

y=3(x2+x+1)

9.

y=4(3x3-2x2+5)+x2+1 (y’=36x2-14x)

(y’=3(2x+1))

x +1
x −1

24. y =

(y’=200 x (x2+1)99)

1
x2 + 1

25. y = 3

⎛
−2 ⎞
⎜ y' =
⎟
⎜
(x − 1) 2 ⎟
⎝
⎠
⎛
− 2x ⎞
⎜ y' = 2
⎟
⎜
(x + 1) 2 ⎟
⎝
⎠

2x 2 − 1
3

x +1
4

⎛
44(2 x − 3) 3 ⎞
⎜ y' =
⎟
⎜
( x + 4) 5 ⎟
⎝ x+4 ⎠
⎝
⎠
2
⎛
x⎞
27. y = x + 1⎜ y' =
⎟
⎜
⎟
x 2 +1 ⎠
⎝
4
3
2
28. y = 2 ⋅ x3 − x 2 + 1 ⋅ (2x 2 + 3) ⎛ y' = 14 x − 12 x + 9 x + 2 x ⎞
⎟
⎜
⎟
⎜
3
2
x − x +1
⎠
⎝
1
1⎞
⎛
29. y=log10x
⎜ y' = log10 e =
⎟
x
x ln10 ⎠
⎝

26. y = ⎛ 2x − 3 ⎞
⎜
⎟

30. y=ln x

(y’=1/x)

3
2
10. y = 2x − 3x + 4x − 5 (y’=3x2-3x+2)
2

31. y=3log2x- 4lnx

11. y=(x2+1) (2x3-4)

(y’=10x4+6x2-8x)

32. y=ln(3x2+4x+5)12. y=1/x

(y’= -1/x2)

13. y=1/x3

(y`= -3/x4)

5

33. y = ln x 2 − 1
34. y = ln(x2 − 1)

6

14. y=1/x

(y’= -5/x )

15. y = 2 + 1 − 3
x3 x 2 x

⎛
3x 2 − 2x − 6 ⎞
⎟
⎜ y' =
⎟
⎜
x4
⎠
⎝
1⎞
⎛
⎜ y' =
⎟
2 x⎠
⎝
2⎞
⎛
⎜ y' = 3 ⎟
3⋅ x ⎠
⎝

16. y =

x

17. y = 3 x 2
18. y = x
5

3

⎛
3
⎜ y' =
⎜
5
5⋅ x2
⎝

⎞
⎟
⎟
⎠

⎛
− 2x 4 + 3x 2 + 4x ⎞
⎜y' = 3
⎟
⎜
⎟
(x 3 + 1) 2
⎝
⎠

35. y=2x
36. y = 2 x

2

37. y = e 2 x

⎛
⎞
x
⎜ y' =
⎟
⎜
(x2 − 1) ln(x2 − 1) ⎟
⎝
⎠

+ x +1
2

−3 x + 5

38. y=e-x

(y’= -1/ex)

39. y = e1 / x
40. y = 10

− 4 + 3log2e ⎞
⎛
⎜ y' =
⎟
x
⎝
⎠
6x + 4 ⎞
⎛
⎜ y' = 2
⎟
3x + 4x + 5 ⎠
⎝
x⎞
⎛
⎜ y' = 2
⎟
x −1⎠
⎝

x

41. y=sen 2x
42. y=sen x2

19. y = 2 ⋅ 3 x 2 − 3x 2 +1
5

4
⎛
⎞
⎜ y' = 3 − 6x ⎟
3⋅ x
⎝
⎠

20. y=(x+1)5

(y’=5(x+1)4)

21. y=(2x2-3x+1)3

(y’=3 (2x2-3x+1)2 (4x-3)) 46. y = cos x

43. y=sen2x
44. y=2 sen x
45. y=sen(x2-2x+1)

ALFONSO GONZÁLEZ
IES FERNANDO DE MENA. DPTO. DE MATEMÁTICAS

I.E.S. "Fernando de Mena"

1
x
48. y=ctg(x2+1)
47.

y = tg

49.

1
3
1
1
y = x3 − x 4 + x 2 −
3
4
2
x

73. y = x 3 x(y’=-3x3+x2+x+1/x2)

⎛
⎞
3
⎜ y' =
⎟
⎜
( x + 2)( x − 1) ⎟
⎝
⎠

76. y=ln(x+1)⋅log(x-1)

⎛
4x ⎞
⎜ y' = 2
⎟
⎜
( x + 1) 2 ⎟
⎝
⎠

log(x − 1) ln(x + 1) ⋅ loge ⎞
⎛
+
⎜ y' =
⎟
x +1
x −1
⎝
⎠

77. y=ln(lnx)

1⎞
⎛
⎜ y' =
⎟
x ln x ⎠
⎝

(y’=1/2)

78. y =

(y’=-2/x2)

51. y=3(x2-x+1) (x2+x-1)

(y’=3(4x3-2x+2))

x2 − 1
x2 + 1

y=

53. y=x/2
54.

123
y =+ 2 + 3 + ln x
xx x

2

1
4
9 1⎞
⎛
⎜ y' = − 2 − 3 − 4 + ⎟
x
x
x
x⎠
⎝

⎛
3 ln ( x + 1) ⎞
⎜ y' =
⎟
⎜
x +1 ⎟
⎝
⎠

2

3

56. y=(2x -1)(x -2)(x +1)
(y’=14x6-25x4+8x3+6x2-10x)
57. y = 1 − x
x2 + 1
3

58. y=ln2x
59. y=ln x2
60. y=(x2+1)(x+2)3
61. y = ln x
x

1
3x − x3 + 2
5

63. y =

x 4 − 2x 2 + 3

64. y = lnx
2
65. y = x + 1
2

x −1

66. y = x2+1
5

4
2
67. y = x − 2 x + 1
5

68. y =

5
x − 2x 2 + 1
4

3
69. y = 3 (x + 1) 3 x + 1

70. y=ln(x-3)

3
ln( x 2 + 1)

1
x+2

79. y = 3

2

55. y=ln3(x+1)

62. y =

x ⋅ ln x

75. y = ln x − 1
x+2

50. y=2/x

52.

74. y =

⎛
7x 2 x ⎞
⎜ y' =
⎟
⎜
2⎟
⎝
⎠
2 + ln x ⎞
⎛
⎜ y' =
⎟
2x⎠
⎝

⎞
⎛
x 4 + 3x2 + 2x
⎟
⎜ y'= −
3
⎜
2
3⎟
2 (x + 1) ⋅ 1−x ⎠
⎝
2 ln x ⎞
⎛
⎜ y' =
⎟
x⎠
⎝

(y’=2/x)
(y’=5x4+24x3+39x2+28x+12)
⎛ y' = 2 − ln x ⎞
⎜
⎟
⎝
2x x ⎠
⎛
− 15x 4 + 3x 2 ⎞
⎜ y' =
⎟
⎜
(3x 5 − x 3 + 2) 2 ⎟
⎝
⎠
3
⎛
2x − 2x ⎞
⎜ y' =
⎟
⎜
⎟
x4 − 2x2 + 3 ⎠
⎝
1
⎞
⎛
⎟
⎜ y' =
2x ln x ⎠
⎝

⎛
− 2x x 2 − 1 ⎞
⎜ y' =
⎟
⎜
( x 2 − 1) 2 ⋅ x 2 + 1 ⎟
⎝
⎠
⎛
2⎞
⎜ y' =
⎟
⎜
⎟
5 ⋅ 5 x3 ⎠
⎝
⎛
4x 3 − 4 x ⎞
⎜ y' =
⎟...