Answers and Explanations 253
Answers
301–400
366.
x
x
x
x
x
−
(
) − + +
(
)
+
(
)
1
7
8
5
2
2
2
6
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
'
'
Applying the quotient rule and the chain rule to f x
x
x
x
( ) =
−
(
)
+
(
)
1
3
2
5
gives you the
following derivative:
=
+
(
)
−
(
) − −
+
(
) +
(
)
+
(
)
(
)
=
x
x
x
x
x
x
x
x
x
x
( )
(
)
(
)
(
)
(
2
5
2
3
2
4
2
5 2
3
1
1 5
2 1
− −
+
(
)
+
(
) − −
+ )
(
+
(
)
=
−
−
+
+
1
3
5
1 2 1
1
7
8
5
2
2
4
2
2
10
2
2
(
)
) (
)
(
)
x
x
x
x
x
x
x
x
x
x
x
( (
)
+
(
)
x
x
2
6
x
f '
367.
−
+
(
)
−
(
)
1
1
1
1 2
3 2
x
x
First rewrite the function using exponential notation:
f x
x
x
x
x
( ) = +
−
=
+
−
( )
1
1
1
1
1 2
Then apply both the chain rule and the quotient rule:
f x
x
x
x
x
x
x
' ( ) =
+
−
( )
−
(
)( )− +
(
)( )
−
(
)








=
+
(
)
−
1
2
1
1
1 1
1 1
1
1
2
1
1 2
2
− −
−
−
(
)
−
−
(
)








=
−
(
)
+
(
)
−
−
(
)







1 2
1 2
2
1 2
1 2
2
1
2
1
1
1
1
1
x
x
x
x
x
 
=
−
+
(
)
−
(
)
1
1
1
1 2
3 2
x
x