Part II: The Answers
252
Answers
301–400
364.
x
x
x
x
11
3
40
4
5
3
3
3 4
+
+
(
)
+
(
)
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
( ) ( )
( ) ( ) ( ) ( )
[
]=
+
'
'
Rewrite the function using exponential notation:
f x
x
x
x
x
( ) =
+
(
) +
=
+
(
) +
(
)
2
3
4
2
3
1 4
1
5
1
5
Then apply the product rule and also use the chain rule on the second factor to get the
derivative:
=
+
(
) + +
(
)
+
(
) ( )
(
)
=
+
(
)
+
−
−
x x
x
x
x
x
x x
( ) ( )
2
5
1 1
4
5
3
5
2
3
1 4
2
3
3 4
2
3
3 4
3
5 5 3
4
1
11
4
3
4
10
5
11
3
40
4
2
2
4
2
3
3 4
3
3
(
) +
+
(
) )
(
=
+
+
+
(
)
=
+
+
(
)
+
x x
x
x
x
x
x
x
x
x 5 5
3 4
(
)
x
f '
365.
6
1
1 7
5
2
3
4
2
5
5
5
x x
x
x
x
−
(
) +
(
)
−
+
(
)
To find the derivative of f x
x
x
( ) =
−
(
) +
(
)
4
3
5
6
1
1 , use the product rule along with the
chain rule and then factor:
f x
x
x
x
x
x
x
x x
' ( ) =
−
(
) ( )
(
) +
(
) + −
(
)
+
(
) ( )
(
)
=
−
3
1 4
1
1 6
1 5
6
4
2
3
5
6
4
3
5
5
4
3
4
1 1
1 2
1 5
1
6
1
1 7
5
2
2
5
5
5
4
3
4
2
5
5
5
(
) +
(
)
+
(
) +
−
(
)
(
)
=
−
(
) +
(
)
−
+
(
)
x
x
x x
x x
x
x
x