Answers and Explanations 239
Answers
301–400
You can apply the quotient rule directly, or you can rewrite the original function as
f x
x
x
x g x
( )
g( )
( )
=
=
[
]
−
3
3
and then apply the product rule to get the following:
= −
(
)[ ]+  
 
= −
+
−
−
x
g x
x
g x
g x
x
g x
x
( )
( )
( )
( )
( )
3
3
4
3
4
3
x
f '
'
'
319.
x sec x tan x + 2 sec
3 
x
Apply the product rule to f
 
(x) = sec x(x + tan x) as follows:
f x
x
x x
x
x
x
x
x
x
x
x
'
sec tan
t an
sec
sec
sec tan
sec tan
s
( ) =
+
(
)+
+
(
)
=
+
+
1
2
2
e ec
sec
sec tan
tan
s ec
sec
sec tan
sec
s
x
x
x
x
x
x
x
x
x
x
x
x
+
=
+
+
(
)
+
=
+
+
3
3
2
3
1
e ec
sec tan
sec
3
3
2
x
x
x
x
x
=
+
320.
2
1
2
x
x
x
x
x
x
+
(
)
−
+
(
)(
)
csc
c sc cot
Apply the product rule to f
 
(x) = (x
2
+ x)csc x to get
=
+
+
+
(
) −
=
+
−
+
(
)
x
x x
x
x
x
x
x x
x
x
( ) (
)csc
( csc cot )
(
)csc
( csc c
2 1
2 1
2
2
o ot )
x
x
f '
321.
4x
2 
(sec x)(3 + x tan x)
Applying the product rule to f
 
(x) = 4x
3
sec x gives you
f x
x
x
x
x
x
x
x
x
x
'( )
s ec
(sec tan )
(sec )(
tan )
= ( )
+
=
+
12
4
4
3
2
3
2
322.
−
−
cot
csc
x
x
x
x
2
3 2
2
1 2
You can apply the quotient rule directly, or you can rewrite the original function as
f x
x
x
x
x
( ) cot
cot
=
=
−
1 2
1 2
and then apply the product rule as follows:
= −
+
−
(
)
= −
−
−
−
x
x x
x
x
x
x
x
( )
cot
c sc
cot
csc
1
2
2
3 2
1 2
2
3 2
2
1 2
x
f '