Matemticas II

CLCULO DE DERIVADAS.

Calcula las derivadas de las siguientes funciones, simplificando al mximo el resultado.

1. y = ln tanx Solucin: y = 12 tanx 1 + tan2x = 1sin2x

2. y = 5 x3 + 2 Solucin: y = 35 5 x3 + 2

4 x2

3. y = e2x + ex 2 + ex2 Solucin: y = 2e2x + 2e2x + 2xex2

4. y = sin 3 3 ln2 + 15 Solucin: y

= 0

5. y = arcsincosx Solucin: y = sinx1 cos2x

= 1

6. y = xx Solucin: y = xxlnx + 1

7. y = x2+ 2x + 1x + 1 Solucin: y

=

2x + 2x + 1

x2 + 2x + 1x + 12

= 1

8. y = sin ln2x + 13 Solucin: y = 3cos ln

12 2x + 13

ln12 2x + 13 2x + 1

9. y = 5x

ln5 Solucin: y= 5x

10. y = x2 cosx + 2x sinx + 2cosx Solucin: y = x2 sinx

11. y = sinxcosxtanx Solucin: y=

cos2xtanx

sin2xtanx sinx

cosxtan2x

1 + tan2x = 2sinxcosx = sin2x

12. y = sinx + cosx2 Solucin: y = 2sinx + cosxcosx sinx = 2cos2x13. y = x2 + 3xesinx Solucin: y = 2x + 3esinx + x2 + 3xcosxesinx

14. y = arccosx2 Solucin: y = 2 x1 x4

15. y = arcsin 1 x Solucin: y = 12 1 x x

16. y = x + 1 xx + 1 + x

Solucin: y =1

2 x+1

12 x

x + 1 + x

x + 1 x

x + 1 + x2

12 x + 1

+1

2 x=

x 1 + x1 + x + x 1 + x x

17. y = 2x + 3sinx Solucin: y = 2x + 3sinx cosx ln2x + 3 + 2 sinx2x + 3

18. y = ex ex

ex + ex Solucin: y= 1 e

x ex 2

ex + ex 2

19. y = ln x2+ 13x Solucin: y

=

12x2 + 1

x 2 x2+ 1x2

=12x

x2 1x2 + 1

20. y = x5 + 5x Solucin: y = 5x4 + 5x ln521. y = cos3x + 3cosx Solucin: y = 3sin3x 3sinx

22. y = arctan 1 x1 + x Solucin: y=

12 1x1+x

11+x

1x1+x2

1 + 1x1+x=

12 1 x2

1

Matemticas II

23. y = 3xex

1 + ln3 Solucin: y= 3xln3 e

x

1 + ln3 + 3x ex1 + ln3 = 3

xex

24. y = x x Solucin: y = x 1x 1x2

lnx + 1x2

= x 1+2xx lnx 1

25. y = sinxcos2x + cosx sin2x Solucin: y = 3cosxcos2x 3sinx sin2x

26. y = x2 1x2

Solucin: y = 2x 2x2 1x3

=2x3

27. y = ln cos5x + 7 Solucin: y = 5 sin5x + 7cos5x + 728. y = 3 x x Solucin: y = 1

2 3 x 32 x

29. y = tan2xcos2x Solucin: y = 2tanxcos2x1 + tan2x 2 tan2xcosx sinx

30. y = 13 2x + 12

Solucin: y = 13 3 2x + 12

4 8x + 4

31. y = arctan x Solucin: y = 12 x 1 + x

32. y = x2+ 2x + 3x Solucin: y

=

2x + 2x

x2 + 2x + 3x2

=x2 3x2

33. y = sin2x + cos2x Solucin: y = 034. y = 5x2+3 Solucin: y = 2 5x2+3x ln5

35. y = lnx3

3 Solucin: y=

1x

36. y = 5 x2 33 Solucin: y = 6

5 5 x2 334 x

2 32x = 6

5 5 x2 32 x

37. y = e3x2 + 2e3 Solucin: y = 6xe3x2

38. y = cos ln x Solucin: y = 12sinln x

x

39. y = 1 x2 arcsinx Solucin: y = 11 x2

arcsinxx + 1

40. y = x tanx Solucin: y = x tanx 1 + tan2x lnx + tanxx41. y = 3sin2x + 2sinx3 Solucin: y = 6sinxcosx + 6cosx3 x2

42. y = x2x Solucin: y = x2x2 lnx + 243. y = tanxcotx Solucin: y = 01 cot2gxg x + g44. y = xe + ex Solucin: y = xe ex + e

x= xe1e + ex

45. y = 2 3 x5 + 72 Solucin: y = 203 3 x5+7 2

2 x5 + 7x4 =20

3 3 x5 + 7x4

46. y = xarctan x2 ln4 + x2 Solucin: y = arctan 12 x +

12

x1 + 14 x

2 2x

4 + x2= arctan 12 x

47. y = lnex2+3x Solucin: y = 2x + 3

48. y = 1 cosx1 + cosx Solucin: y=

12 1cosx1+cosx

sinx1 + cosx +

1 cosx1 + cosx2

sinx = 11 + cosx

49. y = ln 1 + x2 + arctanx Solucin: y = 1x2 + 1

x + 1x2 + 1

=x + 11 + x2

50. y = 2 x Solucin: y = 122 xx

ln2

51. y = x2+ 2x

Solucin: y = 2 x 12x2 + 2 x 3

52. y = arctanx1 + x2

Solucin: y = 1x2 + 12

2 arctanxx2 + 12

x = 1 + 2arctanxx1 + x2 2

53. y = x lnx x Solucin: y = lnx

2

Matemticas II

54. y = arctan1 + x2 Solucin: y = 2 x1 + x2 + 12

55. y = x2ex3 Solucin: y = 2xex3 + 3x4ex3

56. y = esinx + sinex Solucin: y = cosxesinx + cosex ex

57. y = 6x3 x2x

Solucin: y = 3x

3 x2 + 4 x 3

3 x2 2 = 7 6 x

58. y = ln1 + x2 + arctanx Solucin: y = 2x2 + 1

x + 1x2 + 1

=2x + 1x2 + 1

59. y = xcos Solucin: y = 1x2

60. y = x2 1

x + 1 Solucin: y= 1

61. y = sinx + cosx2 sin2x Solucin: y = 2sinx + cosxcosx sinx 2cos2x

62. y = tanx tanx Solucin: y = tantanxx1 + tan2x lntanx + 1 + tan2x = tantanxx 1 + lntanxcos2x

63. y = tanx tan Solucin: y = 0

64. y = ex+ ex2 Solucin: y

=

12 e

x

12 e

x

65. y = x tanx + ln cosx. Solucin: y = tanx + x1 + tan2x sinxcosx =x

cos2x66. y = 3arctan3x Solucin: y = 9

1 + 9x2

67. y = 6x + 33 x2

Solucin: y = 63 x2

23

6x + 33 x2

4 x

68. y = x ln3 Solucin: y = ln369. y = e ln8x Solucin: y = 8

70. y = tanx1 + tan2x

Solucin: y = 1 2 tan2x

1 + tan2x71. y = sin3x4 Solucin: y = 12sin2x4 cosx4 x3

72. y = cos2x1 + tan2x Solucin: y = 2cosx1 + tan2x sinx1 + tan2x1 + tan2x + 2xtanx1 + tan2x

73. y = 1x Solucin: y=

1x2

74. y = 1 + sin2x Solucin: y = 11 + sin2x

cos2x

75. y = e0 Solucin: y = 0

76. y = sinx1 + cosx Solucin: y=

cosx1 + cosx +

sin2x1 + cosx2

=1

1 + cosx

77. y = sin 1 2x Solucin: y = 12cos 1 2x

1 2x 2x ln2

78. y = e1x Solucin: y = 1

x2e 1x

79. y = arcsin x4 Solucin: y=

116 x2

80. y = ex ln sinx Solucin: y = ex lnsinx + ex cosxsinx81. y = tanx 1secx Solucin: y

=

1 + tan2xsecx

tanx 1secx tanx

82. y = 13 tan3x tanx + x Solucin: y = tan2x1 + tan2x tan2x

83. y = x 2x + 2 Solucin: y=

1x + 2

x 2x + 22

84. y = arctan 2x1 x2

Solucin: y =

21 x2

+ 4 x2

1 x2 2

1 + 4 x2

1 x2 2=

21 + x2

3

Matemticas II

85. y = ln x + 1 + x2 Solucin: y =

1 + 1x2 + 1

x

x + x2 + 1=

11 + x2

86. y = lnlnx Solucin: y = 1x lnx87. y = 2 x2 2 x Solucin: y = 2x 2 x 12

2 x22 x

=128x + 5x2 2

2 x

88. y = x lnx Solucin: y = 2 xlnxx lnx

89. y = arcsinx2 Solucin: y = 2 arcsinx1 x2

90. y = 13

arctan x 31 x2

Solucin: y = 13 3

31 x2

+ 2x2 31 x2 2

1 + 3 x2

1 x2 2=

1 + x2x4 + x2 + 1

91. y = lnsin2x Solucin: y = 2 cosxsinx92. y = 10x tanx Solucin: y = 10x tanxtanx + x1 + tan2x ln1093. y = x sinx + cosx Solucin: y = xcosx

94. y = ln tan 4 +x2 Solucin: y

=

12 +

12 tan

2 14 +

12 x

tan 14 +12 x

95. y = earctanx Solucin: y = 1x2 + 1

earctanx

96. y = sin3xcos3x Solucin: y = 3cos23x 3sin23x

97. y = ln 1 + x1 x Solucin: y=

11 x +

1 + x1 x2

1 + x 1 x = 2

x + 1x 1

98. y = x22x

Solucin: y = 12 x2x

2 ln 12 x + 2

99. y = 2 ln sin 4 Solucin: y= 0

100. y = lnxx Solucin: y = lnx + 1

101. y = arctan 1 cosx1 + cosx Solucin: y=

1

2 1 cosx1 + cosx

sinx1 + cosx +

1 cosx1 + cosx2

sinx

1 + 1 cosx1 + cosx=

12

102. y = arccosx2 Solucin: y = 2 x1 x4

103. y = 4x + 23

Solucin: y = 12x + 24

104. y = x lnx1 x + ln1 x Solucin: y=

lnx1 x + x

lnx1 x2

=lnx

1 x2

105. y = 3x + 3 x + x3 Solucin: y = 123x

+1

3 3 x 2+

32 x3

x2

106. y = sincosx Solucin: y = coscosx sinx

107. y = x1x Solucin: y = x 1x 1

x2lnx + 1

x2= x 1+2xx lnx 1

108. y = 12 tan2x Solucin: y = tanx1 + tan2x = sinx

cos3x109. y = x2 + 1x2 1 Solucin: y = 2xx2 1 + 2x2 + 1x = 4x3

110. y = 12x x

Solucin: y = 34 x 5

111. y = cos2x + sin2xcos2x sin2x Solucin: y=

2sin2x + 2cos2xcos2x sin2x

cos2x + sin2xcos2x sin2x2

2sin2x 2cos2x =

4

Matemticas II

42sin2xcos2x 1 =

41 sin4x

112. y = log61 + 2x Solucin: y = 22x + 1ln2 + ln3

113. y = 2sin x Solucin: y = 12 2sin x cos x

xln2

114. y = 1 x1 + x Solucin: y=

12 1x1+x

11 + x

1 x1 + x2

=1

1 + x 1 x2

115. y = ln 1 x1 + x Solucin: y=

121 x 1 + x

11 + x

1 x1 + x2

=1

1 + x1 + x

116. y = tanxa x

Solucin: y = x sinxcosx lnaa x 2 x tanx cos2x

117. y = sin2x + sin4xcos2x + cos4x Solucin: y=

2cos2x + 4cos4xcos2x + cos4x

sin2x + sin4xcos2x + cos4x2

2sin2x 4sin4x =6

4cos32x 3cos2x + 1118. y = ax2 + b4 Solucin: y = 8axax2 + b3

119. y = 2xx 1

Solucin: y = 2x 1

xx 1

3 =x 2

x 13

120. y = x2 + 4x + 4x Solucin: y = x2 + 4x + 4x lnx2 + 4x + 4 + x 2x + 4x2 + 4x + 4

= x + 22x 2xx + 2 + 2 lnx + 2

121. y = 4x arcsinx Solucin: y = 4x ln4arcsinx + 4x

1 x2

122. y = x + ex

x ex Solucin: y=

1 + exx ex

x + exx ex 2

1 ex = 2ex 1 + xx ex 2

123. y = x + lnxx3

Solucin: y = 1 +1x

x3 3 x + lnx

x4=

2x 1 + 3 lnxx4

124. y = 1x + 2 lnx lnxx Solucin: y

=

2x2

+2x +

lnxx2

=2 + 2x + lnx

x2125. y = ex sinx + ex cosx Solucin: y = 2ex cosx = 2ex cosx126. y = xex lnx Solucin: y = ex lnx + xex lnx + ex = ex1 + lnx + x lnx

127. y = xarctanx Solucin: y = arctanx + x1 + x2

=arctanx + arctanxx2 + x

1 + x2

128. y = x + 1xx

Solucin: y = x + 1xx

ln x + 1x + x1 1

x2

x + 1x

129. y = x sinxlnx Solucin: y=

sinxlnx + x

cosxlnx

sinxln2x

130. y = arcsin1 ex Solucin: y = ex

1 1 ex 2=

ex

2ex e2x

131. y = 13 x2 5

Solucin: y = 2x3 3 x2 5

4

132. y = 4x3 + 6x 27 Solucin: y = 74x3 + 6x 2612x2 + 6

133. y = ln arcsinx + arcsin lnx Solucin: y = 11 x2 arcsinx

+1

x 1 ln2x

134. y = x5ex6 Solucin: y = 5x4e

1x6 + 6

x2e

1x6

135. y = 8arcsin 1x Solucin: y = 8

arcsin 1x

x2 1 1x2

ln8

136. y = x cosx Solucin: y = 12 x

cosx x sinx

5

Matemticas II

137. y = 13 x2 lnx x

3

9 Solucin: y=

23 x lnx +

13 x

13 x

2

138. y = exx2 + 3x 9 Solucin: y = exx2 + 3x 9 + ex2x + 3 = exx2 + 5x 6

139. y = ln1 x 11 x Solucin: y=

11 x

11 x2

=x 2

x 12

140. y = arctancosx Solucin: y = sinx1 + cos2x

141. y = 3 sinx Solucin: y = 13sin

23 x

cosx

142. y = sec5x + 2 Solucin: y = 5sec5x + 2 tan5x + 2 = 5sin5x + 2cos25x + 2

143. y = tan35x Solucin: y = 3tan25x5 + 5 tan25x = 15sin25x

cos45x

144. y = x 1 x23

Solucin: y = 3 x 1 x2 2

1 + 11 x2

x =

3 x 1 x2 2 1 x2 + x

1 x2

145. y = axxa Solucin: y = axxa1a + x lna

146. y = ln tan1 x Solucin: y = 1 tan2x 1

tanx 1147. y = 2x + 3

x 14Solucin: y = 2

x 14 4 2x + 3

x 15= 2 3x + 7

x 15

148. y = x 1x + 1x2 + 1 Solucin: y = x + 11 + x2 + x 11 + x2 + 2x 1x + 1x = 4x3

149. y = x sinx Solucin: y = sin1x x 1

x2lnsinx + 1x

cosxsinx

150. y = x2 2x + 325 Solucin: y = 2

5 5 x2 2x + 37 2x 2

151. y = sin ln1 3x Solucin: y = 32cos ln

12 1 3x

ln12 1 3x 1 3x

152. y = 2cos2x2sin2x Solucin: y = 0

153. y = lnx 1 + x2 Solucin: y =1 + x2 + x2

1+x2

x 1 + x2 =

1 + 2x21 + x2 x

154. y = sinx cos 2 x Solucin: y= 0

155. y = ln6x7 x2

14 Solucin: y=

7x

17 x

156. y = 23

arctan 2x + 13

Solucin: y = 43 1 + 13 1 + 2x

2

157. y = x2

4 +2 x2

16 Solucin: y=

12

x

14 +

18 x

158. y = ax + bx c Solucin: y=

ac + bx c2

159. y = tan x2 cotx2 Solucin: y

= 1 + 12 tan

2 12 x +

12 cot

2 12 x =

2sin2x

160. y = sinesinx Solucin: y = cosesinx cosxesinx

6