Answers and Explanations 249
Answers
301–400
Then apply the chain rule to get the derivative:
f x
x x
x
x
x x
' ( ) = −
−
(
)
−
(
)
=
−
−
(
)
−
(
)
−
5
2 1
5 2 1
2
6
2
6
356.
2
1
1
2
2
3
csc
cot
x
x
x












First rewrite the function:
f x
x
x
( ) csc
c sc
=





 =
( )
−
1
2
2
Then apply the chain rule:
f x
x
x
x
x
x
'
c sc
cot
csc
cot
( ) = − ( ) ( )
(
) −
(
)
=












−
−
−
2
2
3
2
2
2
2
1
1
x x
3
357.
5 1 2
2
4
−
(
) +
(
)
cos sin
cos
x
x x
x
Apply the chain rule to f x
x
x
( )
cos
=
+
(
)
2
5 to get
f x
x
x
x
x
x
x x
x
'( )
c os
cos ( sin )
(
cos sin )
cos
=
+
(
) +
−
[
]
=
−
+
(
)
5
1 2
5 1 2
2
4
2
4
358.
π
π
π
cos
s in
x
x
( )+ ( )
(
)
2
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
'
'
To find the derivative of f x
x
x
x
( )
sin( )
cos
s in
=
( )+ ( )
π
π
π
, apply the quotient rule along with
the chain rule:
=
+
(
) −
−
+
x
x
x
x
x
( )
(cos( ) sin( )) cos( )
sin( )(( sin )
(cos(
π
π
π
π
π
π π
πx x
x
x
x
x
x
)) )
cos( ) sin( )
cos ( ) sin ( )
cos( ) sin(
π
π
π
π
π
π
π
π
+
(
)
=
+
(
)
+
2
2
2
x x
x
x
)
cos( ) sin( )
(
)
=
+
(
)
2
2
π
π
π
x
f '