# Derivative and Integral Practice Worksheet (KEY)

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```							                      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

```
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