Answers and Explanations 257
Answers
301–400
379.
x
x
2
5
+
Begin by using properties of logarithms to rewrite the function:
f x
x
x
x
( ) ln
ln
ln
=
+
=
+
(
)
=
+
(
)
2
2
1 2
2
5
5
1
2
5
Then apply the chain rule to find the derivative:
=
+






=
+
x
x
x
x
( )
1
2
1
5
2
5
2
2
( )
x
f '
380.
1
10 6
2
ln
(
) + x
Begin by writing the function using exponential notation:
f x
x
x
x
x
( ) log
log
=
+
+
(
)
=
+ +
(
)
(
)
10
2
10
2
1 2
6
6
Then use the chain rule to find the derivative:
=
+
+





 +
+
(
)
(
)
=
+
+


−
x
x
x
x
x
x
ln
( )
ln
1
10
1
6
1 1
2
6
2
1
10
1
6
2
2
1 2
2
 


 +
+






=
+
+






+
+
+
+






=
1
6
1
10
1
6
6
6
6
2
2
2
2
2
x
x
x
x
x
x
x
x
ln
1 1
10 6
2
(ln ) + x
( )
x
f '
381.
2
6 1 2
3
6 4 3
cos
(ln )(
sin )
cos
(ln )(
sin )
x
x
x
x
− +
−
+
To make the derivative easier to find, use properties of logarithms to break up the
function:
f x
x
x
x
x
( ) log
sin
sin
log
s in
log
s in
=
− +
+
=
− +
−
+
6
6
6
1 2
4 3
1 2
4 3