Chapter 3
126.
Differentiate
() cos
ax
ft x
=
. Assume a is a positive constant.
A)
2
cos sin
cos
a x ax x
x
−
B)
2
cos sin
cos
a x ax x
x
+
C)
sin
a
x
D)
sin
a
x
−
127.
Differentiate
. Assume b is a positive constant.
A)
cos( )
b
b
B)
ln ( ) cos( )
bb
 
C)
1cos( )
bb
b

−
D)
1
cos( )
b
b
−
functions. difficulty: easy section: 3.5
128.
Find the first several derivatives of
( ) sin( )f x ax=
, where a is a constant. Use them to
predict the 15th derivative of
()fx
.
difficulty: medium section: 3.5
129.
If
sin7 cos5y x x=−
, then
dy
dx
=
7cos7 5sin5xx+
.
A)
True
B)
False
functions. difficulty: medium section: 3 review
Chapter 3
130.
If
23
ln( )
x
y x e=
, then
dy
dx
=
23x
x
+
.
A)
True
B)
False
product of functions. difficulty: medium section: 3 review
131.
If
23yx=+
, then
dy
dx
=
2
2
3
x
x+
.
A)
True
B)
False
product of functions. difficulty: medium section: 3 review
132.
The first derivative of
2
cos( )yx=
is
2
2 sin( )xx
.
A)
True
B)
False
functions. difficulty: medium section: 3 review
133.
The first derivative of
ln(4 sin( ))sx=+
is
cos
4 sin
x
x+
.
A)
True
B)
False
functions. difficulty: medium section: 3 review
134.
The first derivative of
2
(3 )tx=+
is
2
2
(3 )
x
x+
.
A)
True
B)
False
product of functions. difficulty: medium section: 3 review
Chapter 3
135.
The first derivative of
2
sin(7 )sx=+
is
2
2 cos(7 )xx+
.
A)
True
B)
False
functions. difficulty: easy section: 3 review
136.
The first derivative of
sin( )x
ye=
is
sin( )
cos( ) x
xe
.
A)
True
B)
False
functions. difficulty: medium section: 3 review
137.
The first derivative of
5
( 2)y x x=+
is
45
5 ( 2) ( 2)x x x+ + +
A)
True
B)
False
product of functions. difficulty: medium section: 3 review
138.
The first derivative of
sin( )s x x=
is
cos( ) sin( )x x x−+
.
A)
True
B)
False
functions. difficulty: easy section: 3 review
139.
The first derivative of
2lnt x x=
is
2 lnx x x+
.
A)
True
B)
False
product of functions. difficulty: medium section: 3 review
140.
The first derivative of
2
4 coss t t=
is
2
4 cos 8 cos( ) sin( )t t t t−
.
A)
True
B)
False
functions. difficulty: medium section: 3 review
Chapter 3
141.
The first derivative of
28w
xw
−
=
is
2
2
8w
w
−
.
A)
True
B)
False
quotient of functions. difficulty: easy section: 3 review
142.
The first derivative of
8
()y x x a=+
is
7
8 ( )x x a+
.
A)
True
B)
False
143.
The first derivative of
2sins x x=
is
2cos 2 sinx x x x+
.
A)
True
B)
False
functions. difficulty: easy section: 3 review
144.
The first derivative of
lnt x x=
is
1 ln x+
.
A)
True
B)
False
product of functions. difficulty: easy section: 3 review
145.
The first derivative of
2
5 sins t t=
is
2
10 cos sin 5sint t t t+
.
A)
True
B)
False
functions. difficulty: medium section: 3 review
146.
The first derivative of
2
( 7) sinx w w=−
is
2 cosww
.
A)
True
B)
False
functions. difficulty: easy section: 3 review
Chapter 3
147.
Differentiating
2
2
2
5
w
yw
−
=−
gives
22
–6
( 5)
w
w−
.
A)
True
B)
False
quotient of functions. difficulty: medium section: 3 review
148.
Differentiating
8
8
cos
z
sz
=
gives
7 8 15 8 8
8 sec 8 sec tanz z z z z+
.
A)
True
B)
False
functions. difficulty: medium section: 3 review
149.
Differentiating
2 sin
2 cos
t
yt
+
=−
gives
2
2cos 2sin 1
(2 cos )
tt
t
−+
−
.
A)
True
B)
False
functions. difficulty: medium section: 3 review
150.
Differentiating
cos( ) 3
sin( ) 2
w
xw
−
=−
gives
2
3cos 2sin 1
(sin 2)
ww
w
−−
−
.
A)
True
B)
False
functions. difficulty: medium section: 3 review
151.
The first derivative of
34yx=+
is
2
3
3
24
x
x+
.
A)
True
B)
False
composition of functions. difficulty: easy section: 3 review
Chapter 3
Page 38
152.
The first derivative of
23
sins z z=+
is
3
2 coszz+
.
A)
True
B)
False
153.
The first derivative of
ln(sin( ) 2)tx=+
is
cos( )
sin( ) 2
x
x+
A)
True
B)
False
154.
The first derivative of
2
7
5
w
xw
−
=+
is
2
22
57
( 5)
ww
w
+−
+
A)
True
B)
False
quotient of functions. difficulty: medium section: 3 review
155.
The first derivative of
3
sin( 7)yx=+
is
23
3 cos( 7)xx+
.
A)
True
B)
False
functions. difficulty: easy section: 3 review
156.
The first derivative of
43
sins z z=+
is
3 2 3
4 3 sinz z z+
.
A)
True
B)
False
functions. difficulty: easy section: 3 review
157.
The first derivative of
cos(sin( ) 5)tx=+
is
cos( )sin(sin( ) 5)xx−+
.
A)
True
B)
False
functions. difficulty: medium section: 3 review
functions. difficulty: easy section: 3 review
Chapter 3
158.
The first derivative of
22
3
w
zw
−
=+
is
2
2
26
( 3)
ww
w
+−
+
.
A)
True
B)
False
quotient of functions. difficulty: medium section: 3 review
159.
The first derivative of
sin x
ye=
is
sin 1
(sin ) x
xe −
.
A)
True
B)
False
160.
The first derivative of
62x
se
−
=
is
62
6x
e−
.
A)
True
B)
False
161.
The first derivative of
x
te=
is
x
xe
.
A)
True
B)
False
composition of functions. difficulty: easy section: 3 review
162.
The first derivative of
( 9)
x
e
ye
+
=
is
( 9)
x
xe
ee +
.
A)
True
B)
False
composition of functions. difficulty: medium section: 3 review
163.
The first derivative of
yx=
is
1
2x
−
.
A)
True
B)
False
functions and sums of power functions. difficulty: easy section: 3 review
Chapter 3
164.
The first derivative of
31
xy y
=+
is
2
2
1
3yy
−
A)
True
B)
False
functions and sums of power functions. difficulty: medium section: 3 review
165.
The first derivative of
22
sin sins t t=+
is
2
2 cos 2sin cost t t t−
.
A)
True
B)
False
166.
The derivative of
32
( ) 8 4 4 16f x x x x= − + +
is
2
24 8 4xx−+
.
A)
True
B)
False
functions and sums of power functions. difficulty: easy section: 3 review
167.
The derivative of
3
26
xx
e+
is
13
2 18
xx
xe
−+
.
A)
True
B)
False
functions and y = ln x. difficulty: easy section: 3 review
168.
The derivative of
10
6
()f x x
x

=+


is
9
1/ 2
2
6 6 1
10 2
xx
xx
−
 
+  − +
 
 
.
A)
True
B)
False
composition of functions. difficulty: hard section: 3 review
169.
The derivative of
2 100
( ) (9 5)f x x=+
is
2 99
1800 (9 5)xx+
.
A)
True
B)
False
composition of functions. difficulty: medium section: 3 review
Chapter 3
170.
The derivative of
4
( ) 7 1f x x
x
= + +
is
2
47
2 7 1
xx
−− +
.
A)
True
B)
False
composition of functions. difficulty: medium section: 3 review
171.
What is the equation of the tangent line to
( ) 2 lnf x x x=
at the point where x = 1?
y = ln x. difficulty: hard section: 3 review
172.
What is the equation of the tangent line to
7
( ) sin
x
P x e x=
at the point where x = 0?
y = ln x. difficulty: hard section: 3 review
173.
Over which of the following intervals is the function
432 9y x x= − +
increasing?
A)
2x
B)
2x
formulas. difficulty: medium section: 3 review
174.
Given the following table, let
( ) ( ( ))h x f g x=
. Find
(3)h
.
x
0
1
2
3
()fx
1
3
5
4
()gx
2
-1
0
1
‘( )fx
3
4
1
-1
‘( )gx
1
3
2
4
Ans:
3
functions. difficulty: easy section: 3 review
Chapter 3
175.
Given the following table, let
( ) ( ( ))h x f g x=
. Find
‘(3)h
.
x
0
1
2
3
()fx
1
3
5
4
()gx
2
-1
0
1
‘( )fx
3
4
1
-1
‘( )gx
1
3
2
4
Ans:
16
functions. difficulty: medium section: 3 review
176.
The vertex form of a parabola is given by
2
( ) ( )f x a x h k= − +
for constants a, h, and k.
At what value of x does
‘( )fx
change sign (i.e. from negative to positive or vice
versa)?
Ans:
difficulty: easy section: 3 review
177.
The vertex form of a parabola is given by
2
( ) ( )f x a x h k= − +
for constants a, h, and k.
For what values of a is the parabola concave up?
difficulty: easy section: 3 review
178.
The amount, W, of fuel used by an aircraft flying at speed v km/min is given, in liters,
by
2400
25Wv
v
=+
. Find the rate of change of W with respect to v when v = 4.
Round to 1 decimal place.
Ans:
175.0
difficulty: medium section: 3 review
179.
Find the derivative of
2
sin(4 ) 5xx
.
difficulty: hard section: 3 review
Chapter 3
Page 43
180.
Find x where the derivative of
2
y ax bx c= + +
is equal to zero.
181.
Find the equation of the tangent line to the graph of
22
( ) 10 + 8 x
p x x e=
at
0x=
(round
to two decimal places).
difficulty: medium section: 3 review
182.
Find the equation of the line tangent to
–2
( ) 7 t
m t e=
at
1t=
, using exact values.
y = ln x. difficulty: hard section: 3 review
183.
Find the equation of the line tangent to
( ) 8 ln(4 )B x x=
at
2x=
, (A) using exact values
and (B) to three decimal places.
Part B:
y= 4 (x – 2)+16.636 or y= 4x+ 8.636
difficulty: medium section: 3 review