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Derivadas
 

Derivadas

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    Derivadas Derivadas Document Transcript

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