Derivative
Practice
Find
the
derivative
of
each
of
the
following
functions.
1.
f (x ) = ln 1+ x x
2.
f (x ) =
2
€
1+
x
1− x 2
3.
f (x ) =
€
3+ x 4
€
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
sin 2x
9.
f (x ) = 3e ( ) 10.
f (x ) = tan4 3x 2 11.
f (x ) = arcsin 3x 12.
f (x ) = arctan ln x
13.
x 2 y + xy 2 = 3x 15.
x sin y + cos 2 y = cos y 16.
cos x − y = xe x (
€
€
€
€
€
€
€
€
2
)
( )
( )
€
( )
( )
5
2
3
( )
€
x2
14.
y + x y = 1+ ye €
Find
the
antiderivative
of
each
of
the
following.
€
€
( )
17.
f (x ) = x 3 − 4 x −3
18.
f (x ) = 4 +5x x
(
4x 3
19.
f x = 4
x +1
2x +3
20.
f (x ) = 2
x +3x +5
()
€
€
)
Answers
2x
1.
f
′ x =
1+ x 2
()
€
€
€
€
€
€
€
() (
2
8.
)
2.
f
′ x = 1+ x
−6x − 4x 3 +2x 5
3.
f
′ x =
2
3+ x 4
4.
f
′ x = −4x −5 tan 3x +3x −4 sec2 3x 5.
f
′ x = e 2x 1+2x 6.
f
′ x = − cos 2x +2sin 2x e −x 1− x 2
7.
f
′ x =
2
2
1+ x
3−2xy − y 2
dy
13.
dx =
x 2 +2xy
()
(
)
()
( )
()
(
( )
)
() ( ( )
()
(
− 32
€
€
€
€
( ))
€
)
€
€
€
14.
dy
dx
=
2xye x −2xy 3
4
2
2
5 y +3x y − e
3
x2
17.
F x = 14 x 4 − 83 x 2 + C 1 5x
18.
F x = 3 +
+C
ln5
x
()
()
2
€
€
€
€
2x
sin 2x
9.
f
′ x = 6cos 2x e ( ) 10.
f
′ x = 24x tan3 3x 2 sec2 3x 2 3x ln3
11.
f
′ x =
2x
1−3
1
12.
f
′ 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
′(x ) =
(x ln x )
(
)
(
(
)
19.
F x = ln x 4 + 1 + C 20.
F x = ln x 2 +3x +5 + C
()
(
()
(
)
)
)