Part II: The Answers
254
Answers
301–400
368.
2
2
2
2
x
x
x
x
cos
c os sin
c os sin sin
( )
(
)
( )
(
)
(
)
( )
(
)
(
)
(
)
To find the derivative of f x
x
( ) =
( )
(
)
(
)
sin sin sin
2
, apply the chain rule repeatedly:
=
( )
(
)
(
)
(
)
( )
(
)
(
) ( )
(
)
=
x
x
x
x
x
( ) cos sin sin
c os sin
c os
( )
cos
2
2
2
2
2
x x
x
x
2
2
2
( )
(
)
( )
(
)
(
)
( )
(
)
(
)
(
)
cos sin
cos sin sin
x
f '
369.
2
1
6
1 2
1 2
1 2
2 3
x
x
x x
+
+
(
)
Rewrite the function using exponential notation:
f x
x
x
x x
( ) =
+
=
+( )
(
)
3
1 2
1 3
Then apply the chain rule repeatedly to get the derivative:
=
+
(
)
+
(
)
=
+
+
(
)
=
+
−
−
−
x x
x
x
x x
x
x
( ) 1
3
1 1
2
1 1
2
3
2
1
6
1 2
2 3
1 2
1 2
1 2
2 3
1 2
1 2 x x x
+
(
)
1 2
2 3
x
f '
370.
1 5
2
9 0
41 47
3
2
6
2
+
(
) + −
(
) − + +
(
)
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
( ) ( )
( ) ( ) ( ) ( )
[
]=
+
'
'
Applying the product rule and chain rule to f
 
(x) = (1 + 5x)
4
(2 + x – x
2
)
7
and then
factoring (factoring can be the tricky part!) gives you the following:
f x
x
x x
x
x x
x
' ( ) =
+
(
) ( )
(
) + −
(
) + +
(
)
+ −
(
) −
(
)
(
)
= +
4 1 5
5 2
1 5
7 2
1 2
1 5
3
2
7
4
2
6
x x
x x
x x
x
x
x
x x
(
) + −
(
)
+ −
(
) + +
(
) −
(
) )
(
= +
(
) + −
(
) −
3
2
6
2
3
2
6
2
2 0 2
7 1 5
1 2
1 5
2
9 90
41 47
2
x
x
+
+
(
)