Chapter 2
The Derivative
Major theorems, figures, and student response questions
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.1.1
(p. 111)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Definition 2.1.1
(p. 111)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.1.2
(p. 112)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.1.7
(p. 116)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.1.11
(p. 117)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Definition 2.2.1
(p. 122)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Finding an Equation for the Tangent Line to y  f  x 
at x  xo
(p. 123)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Definition 2.2.2
(p. 125)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.2.6
(p. 126)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.2.7
(p. 126)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.2.8
(p. 126)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.2.9
(p. 126)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Theorem 2.2.3
(p. 127)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.2.12
(p. 128)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.2.14
(p. 130)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.2.15
(p. 130)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Theorem 2.3.1
(p. 134)
Figure 2.3.1
(p. 134)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.3.2
(p. 135)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Theorem 2.3.2 The Power Rule
(p. 135)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Theorem 2.3.3 Extended Power Rule
(p. 136)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Theorem 2.3.4 Constant Multiple Rule
(p. 136)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Theorem 2.3.5 Sum and Difference Rules
(p. 137)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Theorem 2.4.1 The Product Rule
(p. 143)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Theorem 2.4.2 The Quotient Rule
(p. 144)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Table 2.4.1 Summary of Differentiation Rules
(p. 146)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Formula (3)
(p. 148)
Formula (4)
(p. 148)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Formula (5)
(p. 149)
Formula (6)
(p. 149)
Formula (7)
(p. 149)
Formula (8)
(p. 149)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Theorem 2.6.1 The Chain Rule
(p. 153)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Table 2.6.1
(p. 155)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Definition 2.7.1
(p. 161)
Figure 2.7.1
(p. 161)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.7.4
(p. 164)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.8.1
(p. 168)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
A Strategy for Solving Related Rates Problems
(p. 169)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.9.1
(p. 176)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.9.3
(p. 177)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.9.7
(p. 178)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Derivative and Differential Formulas
(p. 180)
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Select the best answer for the following question.
1. During the first 50 seconds of a missile launch, the
missile is propelled straight up so that in t seconds, it
3
reaches a height of s t   0.6t feet. What is the average
velocity of the missile during the first 50 seconds?
a) 12,500 ft/sec

b) 1,500 ft/sec
n
u
e
s
t
i
o
d) 75,000 ft/sec
1
c) 38,250 ft/sec
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.


2. Which of the following equations has derivative
dy
dx
12x 3  24x  6 ?
a) y  3x 4 12x 2  6x 11
b)
y  36x 2  24
c) y  3x 4 12x 2  6
d) y  12x 4  24x 2  6x 11
Q
u
e
s
t
i
o
n
3

Select the best answer for the following question.
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Select the best answer for the following question.
4 16
x
, find f x.
a)
f x  36x  x
3
b)
f x  36x  x
 54

c)
f x  36x  x

d)
5

f x  36x



3
12
5
5
1
2
1
2
4
34
Q
u
e
s
ti
o
n
4
4
f
x

9x

3. For  
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.



x3  6
4. For y 
, find y.
x3
a) y
2x 3  9x 2  6
x  3
3
2
2
b) y 4x  9x  6
2
x  3
c) y 2x
d) y
2x 3  9x 2  6
x  3
2
Q
u
e
st
io
n
6

Select the best answer for the following question.
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.

Select the best answer for the following question.
5. For f x  3tan x  x cos x , find f x .
a) f x  3sec2 x  sin x

b) f x  3sec x  x sin x  cos x

d) f x  3sec2 x  xsin x  cos x
Q
u
e
st
io
n
7

c) f x  3sec2 x  xsin x  cos x

Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
6. Find an equation of the tangent line to the graph of
y  sin x 1 at x  2 .
a) y  2
b) y
 x 1
c) y  0
Q
u
e
st
io
n
8

Select the best answer for the following question.

d) y  x

Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.




7. For y  5x 14x 
7
2
3
2
 , find
19

a) y 19x 5x 7 14x 2  2
3



19
b) y 5x 14x 
2
3


7
2
c) y 19 35x  28x
6

2
 35x 5  28
6
35x
  28x
18
d) y 19x 5x 14x 
7

18
y.
2
3

18
5
35x
  28
Q
u
e
st
io
n
9

Select the best answer for the following question.
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Select the best answer for the following question.
8. Find dy dx by implicit differentiation:
2
dy
8

3y
12y
a)

dx
3x

b)
dy
8 y

dx x 12y 2
c)
dy
8  3y

dx 3x 12y 2


d) dy 

dx
8  3y
3x 12y 2
3xy 4y 3  8x

Question 1
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Select the best answer for the following question.
9. Assume that oil spilled from a ruptured tanker spreads in a circular
pattern whose radius increases at a constant rate of 3 ft/s. How fast is
the area of the spill increasing when the radius of the spill is 30 ft?
a) 57 ft/sec2
b) 565 ft/sec2
c) 283 ft/sec2
d) 28 ft/sec2
Question 5
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Select the best answer for the following question.
10. The side of a square is measured to be 12 ft, with a
possible error in measurement of ±0.05 ft. Use
differentials to estimate the error in the calculated error.
a) E Area  1.2 ft 2
b) E Area  0.6 ft 2



c) E Area  24 ft 2
d) E Area  7.2 ft 2
Question 5
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
Answers
1. b
6. c
2. a
7. d
3. b
8. c
4. a
9. b
5. d
10. a
Calculus, 10/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.