Part II: The Answers
238
Answers
301–400
313.
x( 2 sin x + x cos x)
Applying the product rule to f
 
(x) = x
2 
sin x gives you
f x
x
x x
x
x
x x
x
'
s in
cos
sin
cos
( ) = ( )
+ (
)
=
+
(
)
2
2
2
314.
(sec ) tan
sec
x
x
x
2
2
+
(
)
Apply the product rule to f
 
(x) = sec x tan x:
f x
x
x
x
x
x
x
x
x
'( ) (sec tan )tan
sec sec
(sec ) tan
sec
=
+
(
)
=
+
(
)
2
2
2
315.
(sec x)(1 + x tan x)
Begin by rewriting the original expression as f x
x
x
x
x
( ) cos
sec
=
=
and then apply the
product rule:
=
+
=
+
x
x x
x
x
x
x
x
( ) ( )sec
sec tan
(sec )(
tan )
1
1
f '
316.
4 (csc x)(1 – x cot x)
Apply the product rule to f
 
(x) = 4x csc x to get
= (
)+ −
(
)
=
−
(
)
x
x
x
x
x
x
x
x
( )
csc
c sc cot
(csc )
cot
4
4
4
1
f '
317.
12
According to the product rule,
( ) ( )
( ) ( ) ( ) ( )
fg x
f x g x f x g x
=
+
'
'
'
'
To find (fg) ' (4), enter the numbers and solve:
( ) ( )
( ) ( ) ( ) ( )
( )( ) ( )( )
fg
f
g
f
g
=
+
=
− +
=
4
4
4
4
4
2 6
3 8
12
'
'
'
318.
−
( ) + ( )
3
4
3
g x
x
g x
x
'
Recall that the product rule states
d
dx
f x g x
g x f x g x
( ) ( )
( ) ( ) ( ) ( )
[
]=
+
x
f '
'