Derivative and Integral Practice Worksheet (KEY)

							                      Derivative and Integral Practice Worksheet (KEY)

        dy                                              8.   y = 46
Find       .
        dx                                                    dy
                                                                 =0
   1.      y = ln x                                           dx
           dy 1
             =                                          9.   y = e x ln 3           (OR )    y = e x ln 3
           dx x
                                                                                             y = eln 3
                                                                                                       x
                                                             dy
                                                                = e x ln 3 ⋅ ln 3
   2.      y = −4sin x                                       dx                              y = 3x
                                                             dy
                                                                = eln 3 ⋅ ln 3
                                                                        x
           dy                                                                                ln y = x ln 3
              = −4 cos x                                     dx
           dx                                                                                1 dy
                                                             dy                                    = ln 3
   3.      y = ln ( − cos x )                                   = 3x ⋅ ln 3                   y dx
                                                             dx
           dy    1                                                                            dy
              =       ⋅ sin x                                                                    = 3x ⋅ ln 3
           dx − cos x                                                                         dx
           dy
              = − tan x
           dx                                           10. y = tan 2 (3x )
   4.     y = 4x                                             dy
                                                                = 2 tan (3 x ) ⋅ sec 2 (3 x ) ⋅ 3
          ln y = x ln 4                                      dx
                                                              dy
          1 dy
                = ln 4                                           = 6 tan (3x ) ⋅ sec2 (3x )
           y dx                                               dx
           dy                                           11. cos y = e x
              = ( ln 4 ) 4 x
           dx                                                          dy
                                                             − sin y      = ex
   5.
           y=
                (3x −1)(5x +12)                                        dx

                    4x −8                                     dy   ex
                                                                 =
          ln y = ln(3x −1) + ln(5x +12) − ln( 4x −8)          dx − sin y
           1 dy   3      5     4                        12. y = e x5
                =    +      −
           y dx 3x −1 5x +12 4x −8
                                                              dy
                                                                 = ex ⋅ 5x4
                                                                     5

           dy  3        5     1  (3x −1)(5x +12)          dx
             =      +      −                     
           dx  3x −1 5x +12 x − 2     4x −8      
                                                        13. y = x e5
   6.      y = tan ( ln x )                                   dy
                                                                 = e5 ⋅ x e −1
                                                                           5


           dy                 1                             dx
              = sec2 ( ln x )  
           dx                  x                      14. y = e5 x
   7.          1                                              dy
           y=                                                    = 5e5 x
               x                                              dx
           y = x −1
           dy              1
              = − x −2 = − 2
           dx             x
15. y = e x (3 x − 2 )                                  21. y = x 5 ln 4 x
                                                                                                 4 
         = (3 x − 2 ) ( e x ) + ( e x ) (3 )                     = ( ln 4 x ) (5 x 4 ) + ( x5 )  
      dy                                                      dy
      dx                                                      dx                                 4x 
      dy                                                      dy
         = 3x ⋅ e x + e x                                        = 5 x 4 ln 4 x + x 4
      dx                                                      dx
16. y = tan e x                                         22.        x
                                                              y = ∫ cos 4t dt
         = e x ⋅ sec 2 ( e x )
      dy                                                           2

      dx                                                      g ( x ) = ∫ cos 4t dt
17. y = 1000 (1.03x )                                         g ′ ( x ) = cos 4 x
     ln y = ln1000 + x ln1.03                                 y = g ( x ) − g (2)
     1 dy                                                     dy
           = ln1.03                                              = g′ ( x) − 0
      y dx                                                    dx
                                                              dy
         = ( ln1.03)(1000 ) (1.03x )
      dy                                                         = cos 4 x
      dx                                                      dx

18. y = (sin x )3 x                                     23.        x2
                                                              y = ∫ sec3 t dt
     ln y = 3x ⋅ ln (sin x )                                       3


                                         cos x              g ( x ) = ∫ sec3 t dt
           = (ln (sin x )) (3) + (3 x ) 
      1 dy
                                                
      y dx                               sin x              g ′ ( x ) = sec3 x
                                                              y = g ( x 2 ) − g (3)
      dy
      dx
         = (3ln (sin x ) + 3 x cot x ) (sin x )(
                                               3x
                                                    )
                                                                 = g ′ ( x2 ) ⋅ 2 x − 0
                                                              dy
19. y = ln ( tan x )                                          dx
      dy
         =
           1
      dx tan x
               ⋅ sec 2 x
                                                              dy
                                                              dx
                                                                           (
                                                                 = sec3 ( x 2 ) ( 2 x )       )
      dy cos x      1
         =     ⋅                                        24.        sin x
                                                              y=    ∫
                                                                               3
      dx sin x cos 2 x                                                             t dt
                                                                       0
      dy       1
         =             = csc x sec x                          g ( x ) = ∫ 3 t dt
      dx sin x cos x
20. y = ( 2 x − 5 )−1                                         g′ ( x) = 3 x
                                                              y = g (sin x ) − g ( 0 )
      dy
         = − (2 x − 5) (2 )
                      −2
                                                              dy
      dx                                                         = g ′ (sin x ) ⋅ cos x − 0
                                                              dx
      dy       −2
         =
      dx ( 2 x − 5)2                                          dy
                                                              dx
                                                                           (
                                                                 = 3 sin x cos x          )
   25.           ln x                 33.
                                            ∫ ln (e ) dx
                                                    5x
         y=
                   x
                                            = ∫ 5 x dx
                   ( x)
                  1
                     − ( ln x )(1)
                       
         dy
            =   x                             5 2
                                            =     x +C
         dx           x2                        2
         dy 1 − ln x
            =
         dx    x2
                                      34.
                                            ∫ 5dx
                                            = 5x + C
Integrate.
   26.     5                          35.
                                            ∫ cot 3x dx
         ∫ x dx                               cos 3 x
                                            =∫        dx
         = 5ln x + C                          sin 3 x
                                             1
   27.       1                              = ln sin 3x + C
         ∫ 3x dx                             3

          1
         = ln x + C
          3
          1
         = ln 3x + C
          3
   28.        x3
         ∫ 5 − 3x 4 dx
            −1
         = ln 5 − 3 x 4 + C
            12
   29.       sin x
         ∫ cos x dx
         = − ln cos x + C

   30.
         ∫ tan x dx
                 sin x
         =∫            dx
                 cos x
         = − ln cos x + C

   31.
         ∫e
              4x
                   dx
              1 4x
         =      e +C
              4
   32.
         ∫x e
              3 x4
                     dx
              1 x4
         =      e +C
              4