Part II: The Answers
260
Answers
301–400
You can now further simplify by using the change of base formula to write log
ln
ln
5
5
x
x
=
so that you have
2
1
6 5
5
1
2
1
6 5
5
5
4
(
) ( ln ) log
(ln )
(
) ( ln ) ln
ln
(l
x
x
x
x x
x
x
x
+ −
(
)
+
(
)
=
+ −
( )
n n )
(
)
ln
(ln )
5
1
2
1 6
5
1
4
4
x x
x
x x
x x
+
(
)
=
+ −
+
(
)
386.
x
x
x
x x
x
tan
sec
ln
tan
2
1
(
) + ( ) ( )

 

 
Begin by rewriting the function and taking the natural logarithm of each side:
y x
y
x
y
x
x
x
x
=
= ( )
=
tan
tan
ln( ) ln
ln( ) (tan )ln
Take the derivative of each side with respect to x:
1
1
2
y
dy
dx
x
x
x x
= (
) + ( ) ( )
sec
ln
tan
Then multiply both sides by y:
dy
dx
y
x
x
x x
= (
) +
( )

 

 
sec
ln
(tan )
2
1
Finally, replacing y with x
tan x
gives you the answer:
dy
dx
x
x
x
x x
x
=
(
) +
( )

 

 
tan
sec
ln
(tan )
2
1
387.
ln
sin
ln ln
cos
ln
cos
x
x
x
x
x x
x
( )
−
(
) ( )
 
  +

 

 
Begin by rewriting the function and taking the natural logarithm of each side:
y
x
y
x
y
x
x
x
x
= ( )
=
( )




=
[
]
ln
ln( ) ln ln
ln
(cos ) ln(ln )
cos
cos
Then take the derivative of each side with respect to x:
1
1
1
y
dy
dx
x
x
x
x x
= −
[
]+
( )






( sin ) ln(ln ) (cos ) ln
Multiplying both sides by y produces
dy
dx
y
x
x
x
x x
=
−
[
]+
( )












( sin ) ln(ln ) (cos ) ln
1 1