Ejercicios De Derivadas

100 DERIVADAS RESUELTAS

Hallar las derivadas simplificadas de las siguientes funciones:
1. 2. 3. 4. 5. 6. 7. y=3 y=x y=5x y=x y=x4+x3+x2+x+1 y= 4x -x +3x -7
4 3 2

(y’=0) (y’=1) (y’=5) (y’=1) (y’=4x3+3x2+2x+1) (y’=16x -3x +6x)
3 2

22. y=(x2+1)100 23. y = 24. y =

(y’=200 x (x2+1)99)
⎛ −2 ⎞ ⎜ y' = ⎟ ⎜ (x − 1) 2 ⎟ ⎝ ⎠ ⎛ − 2x ⎞ ⎜ y' = 2 ⎟ ⎜ (x + 1) 2 ⎟ ⎝ ⎠

x +1 x −1
1 x2 + 1
2x 2− 1 x +1
4

25. y = 3

3

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

1 1 1 y = − x5 + 4x4 − x3 + x 2 − 3 5 6 2
1 2 ⎛ ⎞ 4 3 ⎜ y' = − x + 16x − x + x ⎟ 2 ⎝ ⎠

8. 9.

y=3(x2+x+1)

(y’=3(2x+1))

y=4(3x3-2x2+5)+x2+1 (y’=36x2-14x)

⎛ 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 ⎞ ⎜ ⎟

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

30. y=ln x 31. y=3log2x- 4lnx 32. y=ln(3x2+4x+5) 33. y = ln x 2 − 1 34. y = ln(x2 − 1) 35. y=2x 36. y = 2 x 38. y=e-x 39. y = e1 / x 40. y = 10
x
2

(y’=1/x)
− 4 + 3log2e ⎞ ⎛ ⎜ y' = ⎟ x ⎝ ⎠ 6x + 4 ⎞ ⎛ ⎜ y' = 2⎟ 3x + 4x + 5 ⎠ ⎝ x ⎞ ⎛ ⎜ y' = 2 ⎟ x −1⎠ ⎝
⎛ ⎞ x ⎜ y' = ⎟ ⎜ (x2 − 1) ln(x2 − 1) ⎟ ⎝ ⎠

11. y=(x2+1) (2x3-4) 12. y=1/x 13. y=1/x3 14. y=1/x
5

(y’=10x4+6x2-8x) (y’= -1/x2) (y`= -3/x4) (y’= -5/x )
⎛ 3x 2 − 2x − 6 ⎞ ⎟ ⎜ y' = ⎟ ⎜ x4 ⎠ ⎝ 1 ⎞ ⎛ ⎜ y' = ⎟ 2 x⎠ ⎝ 2 ⎞ ⎛ ⎜ y' = 3 ⎟ 3⋅ x ⎠ ⎝
⎛ 3 ⎜ y' = ⎜ 5 5⋅ x2 ⎝ ⎞ ⎟ ⎟ ⎠
6

15. y = 2 + 1 − 3 x3 x 2 x 16. y =

+ x +1
2

x

37. y = e 2 x−3 x + 5

(y’= -1/ex)

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

41. y=sen 2x 42. y=sen x2 43. y=sen2x 44. y=2 sen x 45. y=sen(x2-2x+1)

19. y = 2 ⋅ 3 x 2 − 3x 2 + 1 5 20. y=(x+1)5 21. y=(2x2-3x+1)3

4 ⎛ ⎞ ⎜ y' = 3 − 6x ⎟ 3⋅ x ⎝ ⎠

(y’=5(x+1)4)

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

I.E.S. "Fernando de Mena"

ALFONSO GONZÁLEZ IES FERNANDO DE MENA. DPTO. DE MATEMÁTICAS

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

y = tg

73. y = x 3 x 74. y =

49.

1 3 1 1 y = x3 − x 4 + x 2 − 3 4 2 x

(y’=-3x3+x2+x+1/x2)

x ⋅ ln x

⎛ 7x 2 x ⎞ ⎜ y' = ⎟ ⎜ 2 ⎟ ⎝ ⎠ 2 + ln x ⎞ ⎛ ⎜ y' = ⎟ 2 x ⎠ ⎝
⎛ ⎞ 3 ⎜ y' = ⎟ ⎜ ( x + 2)( x − 1) ⎟ ⎝ ⎠

50. y=2/x 51. y=3(x2-x+1) (x2+x-1) 52.
y= x2 − 1 x2 + 1

(y’=-2/x2) (y’=3(4x3-2x+2))
⎛ 4x ⎞ ⎜ y' = 2 ⎟ ⎜ ( x + 1) 2 ⎟ ⎝ ⎠

75. y = ln x − 1 x+2 76.y=ln(x+1)⋅log(x-1) 77. y=ln(lnx) 78. y =
3 ln( x 2 + 1)

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

53. y=x/2 54.
1 2 3 y = + 2 + 3 + ln x x x x

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

55. y=ln3(x+1)
2 2 3

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

79. y = 3

1 x+2

⎞ ⎛ 6x ⎟ ⎜ y' = − 2 2 2 ⎜ (x + 1) ⋅ ln (x + 1) ⎟ ⎠ ⎝ ⎛ ⎞ 1 ⎜ y' = − ⎟4 ⎟ 3 ⎜ 3 ⋅ ( x + 2) ⎠ ⎝
⎛ 2x 3 + 3x 2 − 5 ⎞ ⎜ y' = 3 ⎟ ⎜ (x + 1) 2 ⎟ ⎝ ⎠ ⎛ 42x − 1 + ln(3x 2 − 5) ⎜ y' = ⎜ ln2 (3x 2 − 5) ⎝

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

2 80. y = 3 (x − 1) (x + 2) x +1

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

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

2 81.y = 7 3x − 5 ln(3x 2 − 5)

[

]⎞ ⎟
⎟ ⎠

82. y = e 83. y=x⋅ex 84. y = e 85. y =
x

x2

(y' = e

x2

⋅ 2x

)

(y’=(x+1)⋅ex)
⎛ e x ( x − 1) ⎞ ⎜ y' = ⎟ ⎜ x2 ⎟ ⎝ ⎠
⎛ lnx − 2 ⎞ ⎜ y' = ⎟ ⎜ 2 x ln 2 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 ⎠ ⎝

x

x ln x
x+3

62. y = 63. y =

1 3x − x3 + 2
5

86. y = 2 x + 4

⎞ ⎛ x+4 ⎟ ⎜ y' = ⎜ (x + 3) x + 3 ⎟ ⎠ ⎝
⎛ ⎜ y' = ⎜ ⎝ ⎞ ⎟ - x 4 + 8x 2 - 15 ⎟ ⎠ 2x

x 4 − 2x 2 + 3

2 87. y = arc sen (x − 4)

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

88. y = arc cos 89. y = arc tg

x −1

66. y = x2 +1
5
4 2 67. y = x − 2 x + 1 5

⎛ − 2x x 2 − 1 ⎞ ⎜ y' = ⎟ ⎜ ( x 2 −...