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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 )

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Derivadas

  • 1. 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
  • 2. 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 )
  • 3. 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
  • 4. 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
  • 5. 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
  • 6. 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)
  • 7. ejerciciosyexamenes.com 1 - senx sen( ecosx ) - l n cos x cos( ecosx ) e cosx (-senx) 100. y′ = cosx 2 cosx sen 2 ( ecosx )