Part II: The Answers
242
Answers
301–400
330.
−
+
20
4
6
5
g x
x
g
x
( )
x
( )
'
You can use the quotient rule directly, or you can rewrite the original expression as
f x
g x
x
x g x
( )
( )
( )
=
=
−
4
4
5
5
and then apply the product rule as follows:
= −
+
=
−
+
−
−
x g x
x
g x
x
x
( )
( )
20
4
20
4
6
5
6
5
x
( )
f '
g x
( )
'
g x
( )
'
331.
sin g
sin g
cos g
x
x h x
x
x h x
x
x h x
(
) ( ) ( )+ (
) ( ) ( )+ (
) ( ) ( )
′
′
To find the derivative of f x
x h x
x
( ) g( ) ( ) sin
= [
]
, use the product rule within the
product rule:
=
+
 
 
+ [
]
=
h x
x h x
x
x h x
x
x
g
) g( ) ( ) sin
g( ) ( ) cos
(sin )g ( x x h x
x
x h x
x
x h x
) ( ) (sin )g( ) ( ) (cos )g( ) ( )
+
+
x
( )
f '
x
( )
'
(
'
'
'
332.
5
3
4
2
(
)
x +
Recall that the quotient rule states
d
dx
f x
g x
g x f x f x g x
g x
( )
( )
( ) ( ) ( ) ( )
( )





 =
−
[
]
2
'
'
Apply the quotient rule to f x
x
x
( ) =
+
+
2 1
3
4
:
=
+
−
+
+
=
+
x
x
x
x
(
)( ) (
)( )
(
)
(
)
3
4 2
2 1 3
3
4
5
3
4
2
2
x
( )
f '
333.
10 2
5
2
2
2
−
+
(
)
x
x
Apply the quotient rule to f x
x
x
( ) = +
2
5
2
to get the derivative:
x
x
x
x
x
x
( ) ( )( )
=
+
(
) −
+
(
)
=
−
+
(
)
5
2
2
2
5
10 2
5
2
2
2
2
2
2
x
( )
f '
334.
cos
sin
cos
x
x
x
−
+
+
(
)
1
1
2
Apply the quotient rule to f x
x
x
( ) sin
cos
=
−
+
1
1
: