Part II: The Answers
256
Answers
301–400
375.
6
x
Recall that the chain rule states
d
dx
f g x
f g x g x
( )
( ) ( )
(
)= ' (
) '
Using the chain rule on f
 
(x) = H(x
3
) gives you the following:
=
' ( )



 ( )
H x
x
3
2
3
f x
' ( )
Because H x
x
' ( ) = 2 , you know that that H x
x
'
3
3
2
( ) = , so the derivative becomes
=
' ( )



 ( )
=
( )
=
H x
x
x
x
x
3
2
3
2
3
2 3
6
( )
x
f '
376.
80
Because the chain rule gives you
= ' (
)
 
  '
x
f g x
g x
( )
( )
( )
F '
, it follows that
= ' (
)
 
  '
= '
 
 
=
=
f g
g
f
( )
( )
( )
( ) ( )
( )( )
4
4
4
6
8
10 8
80
F '
377.
2
x
Use properties of logarithms to rewrite the function:
f x
x
x
( ) ln
ln
=
=
2
2
Then take the derivative:
f x
x x
'( ) = =
2 1 2
378.
4
3
ln x
x
( )
To find the derivative of f
 
(x) = (ln x)
4
, apply the chain rule:
=
( )
=
x
x
x
x
(ln )
(ln )
4
1
4
3
3
( )
x
f '