Answers and Explanations 259
Answers
301–400
384.
1
2
2
+ x
x
To find the derivative of f x
x
x
x
x
( )
ln
= −
+ +
+
+
(
)
2
2
1
1 , apply the quotient rule and
chain rule to the first term and apply the chain rule to the second term:
=
−
+
(
)
+
+
(
) )
(
+
+
+
(
)
+
+
−
x
x
x
x
x
x x
x
( )
( )
1
2
1
2
1
1
1
1
1 1
2
2
1 2
2
1 2
2
2
1 2
2
1 1
2
1
1
1
1
1
1
1 2
2
1 2
2
2
2
2
1 2
2
(
)
(
)
=
+
(
) − + +
(
)



 +
+
+
(
)
+
+
−
−
( )
x
x
x
x
x
x x
x
x
( (
)



 



 
=
+
(
)
+
+
+
(
)



 



 
+
(
) +
1 2
2
2
1 2
2
1 2
2
1 2
1
1
1
1
1
x x
x x
x
x x
x
x x
x
x
x x
2
1 2
2
2
1 2
2
1 2
2
2
2
1
1
1
1
1
1
1
1
+
(
)



 



 
=
+
(
)
+
+
(
)
=
+
(
)
+
(
)
2 2
2
2
1
=
+ x
x
( )
x
f '
385.
2
1 6
5
1
4
x
x x
x x
+
(
)−
( )
+
(
)
ln
ln
Begin by using properties of logarithms to rewrite the function:
f x
x
x
x
x
( )
log
log
=
+
(
)
=
+
(
)
5
2
3
5
3
1
2
1
Then apply the quotient rule and the chain rule:
=
+
− (
)
+
(
)
+
=
+
x
x
x
x
x
x
x
(
) ( ) (ln )
log
(
)
(
)
(
)
ln
1 2
1
5
2
3
1
1
1
2
5
3
5
2
6
2
( (
)
log
(
)
(
)
ln
log
(
)
(
) (
x
x
x
x
x
x
x
x
+ −

 

 
+
=
+ −
+
=
+ −
1 6
1
2
1
5
6
1
2
1
5
6
5
4
6 6 5
5
1
5
4
ln ) log
(ln ) (
)
x
x
x x
(
)
+
( )
x
f '