Derivative Practice Find the derivative of each of the following functions. 1. f (x ) = ln 1+ x x
2. f (x ) =
2
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1+
x
1− x 2
3. f (x ) =
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3+ x 4
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4. f (x ) = x −4 tan 3x 5. f (x ) = xe 2x €
6. f (x ) = e −x cos 2x €
dy
Use implicit differentiation to find dx of the following. €
x
7. f (x ) =
1+ x 2
e 2x
8. f (x ) =
x ln x
13. x 2 y + xy 2 = 3x 15. x sin y + cos 2 y = cos y 16. cos x − y = xe x (
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2
)
( )
( )
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14. y 5 + x 2 y 3 = 1+ ye x €
Find the antiderivative of each of the following. €
( )
17. f (x ) = x − 4 x −3
18. f (x ) = 4 +5x x
(
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)
4x 3
19. f x = 4
x +1
2x +3
20. f (x ) = 2
x +3x +5
()
3
€
( )
( )
€
( )
2
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sin 2x
9. f (x ) = 3e ( ) 10. f (x ) = tan4 3x 2 11. f (x ) = arcsin 3x 12. f (x ) = arctan ln x
Answers 2x
1. f ʹ′ x =
1+ x 2
()
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() (
2
8. )
2. f ʹ′ x = 1+ x
−6x − 4x 3 +2x 5
3. f ʹ′ x =
2
3+ x 4
4. f ʹ′ x = −4x −5 tan 3x +3x −4 sec2 3x 5. f ʹ′ x = e 2x 1+2x 6. f ʹ′ x = − cos 2x +2sin 2x e −x 1− x 2
7. f ʹ′ x =
2
2
1+ x
3−2xy − y 2
dy
13. dx =
2
x
+2xy
()
(
)
()
( )
()
(
( )
)
() ( ( )
()
(
− 32
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€
€
( ))
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)
€
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2xye x −2xy 3
dy
14. dx
=
4
2 2
x2
5
y
+3x
y
−
e
3
17. F x = 14 x 4 − 83 x 2 + C 1 5x
18. F x = 3 +
+ C ln5
x
()
()
2
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€
€
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2x
sin 2x
9. f ʹ′ x = 6cos 2x e ( ) 10. f ʹ′ x = 24x tan3 3x 2 sec2 3x 2 3x ln3
11. f ʹ′ x =
2x
1−3
1
12. f ʹ′ x =
2 ⎞
⎛
x ⎜ 1+ ln x ⎟
⎝
⎠
()
( )
( ) ( )
()
()
()
( )
sin y
dy
15. dx
=
2sin 2 y − x cos y − sin y
e x x + 1 + sin x − y
dy
16. dx =
sin x − y
( )
2
€
(2x ln x − ln x − 1)e
f ʹ′(x ) =
(x ln x )
(
)
(
(
)
19. F x = ln x 4 + 1 + C 20. F x = ln x 2 +3x +5 + C
()
(
()
(
)
)
)