Part II: The Answers
250
Answers
301–400
359.
−
+
(
) (
)
sin
cos sin sin
x x
x
x
x
Recall that the chain rule states
d
dx
f g x
f g x g x
( )
( ) ( )
(
)= (
)
'
'
and that the product rule states
d
dx
f x g x
f x g x f x g x
( ) ( )
( ) ( ) ( ) ( )
[
]=
+
'
'
To find the derivative of f
 
(x) = cos(x sin x), apply the chain rule and the product rule:
f x
x
x
x x
x
x x
x
x
x
'
s in sin
sin
cos
sin
cos sin sin
( ) = − (
)
 
 
+
[
]
= −
+
(
) (
1
1
) )
360.
sin
c os
x
x
x
3
3
3
1
3
+
Rewrite the function using exponential notation:
f x
x
x
x
x
( )
sin
sin
=
=
( )
3
1 3
Now apply the product rule, being careful to use the chain rule when taking the
derivative of the second factor:
f x
x
x
x
x
x
x
x
'
s in
cos
sin
c os
( ) =
( ) +
( )
(
)(
)
=
( ) + ( )
−
1
1
3
1
3
1 3
1 3
2 3
1 3
1 3
1 3
( ( )
(
)
=
+
sin
c os
x
x
x
3
3
3
1
3
361.
3
3 2
3 2
−
−
(
)
x
x
Recall that the chain rule states
d
dx
f g x
f g x g x
( )
( ) ( )
(
)= (
)
'
'
and that the quotient rule states
d
dx
f x
g x
g x f x f x g x
g x
( )
( )
( ) ( ) ( ) ( )
( )





 =
−
[
]
2
'
'
Rewrite the function using exponential notation:
f x
x
x
x
x
( ) =
−
=
−
(
)
3 2
3 2
1 2