Derivadas ejercicios propuestos

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