The document covers fundamental concepts of differentiation, including the definition of a derivative, various differentiation techniques such as the power rule, product rule, and the sum rule. It also discusses the instantaneous rate of change, average rate of change, and provides examples related to population growth and economic indicators like Gross National Product (GNP). Moreover, it introduces the concepts of marginal cost and revenue, along with the second derivative test for identifying concavity and critical points in functions.

1. Chapter Two
Differentiation:
Basic Concepts
ដដដ
ដដដ

2. Definition
For the function y f(x), the derivative
of f with respect to x is
The Derivative
f (x x) f (x)
f x lim
x 0 x
if the limit exists.
2

3. The Power Rule
Techniques of Differentiation
For any number n,
d n n 1
x nx
dx
Example 1
Differentiate (find the derivative of) each
of the following functions:
27 1 1
y x y 27
y x y
x x
3

4. The Derivative of a constant
Techniques of Differentiation
For any constant C,
d
C 0
dx
4

5. The Constant Multiple Rule
Techniques of Differentiation
For any constant C,
d df
Cf C Cf x
dx dx
Example 2
5
Differentiate the function y 3x
d 5 d 5 4 4
3x 3 x 3 5x 15 x
dx dx
5

6. The Sum Rule
Techniques of Differentiation
d df dg
f g f x g x
Example 3 dx
dx dx
d 2 5 d 2 d 5
x 3x x 3x
dx dx dx
4
2 x 15 x
6

7. The Product Rule
Techniques of Differentiation
d df dg
fg g f
dx dx dx
f x g x g x f x
Example 4
d d 2 d
x 2 3x 1 3x 1 x 2
x 3x 1
dx dx dx
3 x 1 2x x 2 3
9x 2 2x
7

8. The Derivative of a Quotient
Techniques of Differentiation
df dg
g f
d f dx dx
2
dx g g
Example 5
Differentiate the rational function
2
x 2 x 21
y
x 3
8

9. The Derivative of a Quotient
Techniques of Differentiation
Example 5
d d
x 3 x2 2 x 21 x 2
2 x 21 x 3
dy dx dx
2
dx x 3
x 3 2x 2 x2 2 x 21
2
x 3
2x 2 4 x 6 x 2 2 x 21 x2 6 x 15
2 2
x 3 x 3
9

10. For the function y f(x), the change in x
Average Rate of Change
is Δx, the change in y is Δy and
Δy f(x Δx) f(x)
 The average rate of change (ARC) of y
with respect to x is
ARC Δy/ Δx
Or
f x x f x
ARC
x
10

11. Instantaneous Rate of Change
When Δx approaches zero, the
average rate of change becomes
instantaneous rate of change (IRC).
It is the derivative of the function f at
any point x.
IRC=f’(x)=dy/dx
11

12. Instantaneous Rate of Change Example1
It is estimated that x months from now, the
population of a certain community will be
P(x) x2 20x 8,000
a) At what rate will the population be
changing with respect to time 15
months from now?
b) By how much will the population
actually change during the 16th month?
12

13. Instantaneous Rate of Change Example1
a) The rate of change of the population with
respect to time is the derivative of the
population function. That is,
Rate of Change P’(x) 2x 20
The rate of change of the population 15
months from now will be
Rate of Change 2 15 20
Rate of Change 50 people /month
13

14. Instantaneous Rate of Change Example1
b) The actual change in the population
during the 16th month is the difference
between the population at the end of
16 month and the population at the
end of 15 months. That is,
DP P(16) P(15)
DP 8576 8525 51 people/month
14

15. Percentage Rate of Change
For y f(x), the percentage rate of
change of y with respect to x is defined
by
f x
PRC 100
f x
15

16. Percentage Rate of Change Example 2
The gross national product (GNP) of a
certain country was N(t)=t2+5t+106 billion
dollars years after 1980.
a) At what rate was the GNP changing
with respect to time in 1988?
b) At what percentage rate was the
GNP changing with respect to time in
1988?
16

17. Percentage Rate of Change Example 2
a) The rate of change of GNP is the
derivative of N(t) when t=8 (in 1988)
N’(t)=2t+5
N’(8)=2(8)+5=21 billion $/year
b) The percentage rate of change of the
GNP in 1988 was
PRC = 100 [N’(8)/N(8)]
PRC = 100 (21/210)=10%/year
17

18. Approximation by Differentials
rate of change
Change change
of y with
in y in x
respect to x
18

19. Approximation by Differentials
If y=f(x), and Δx a small change in x
than the corresponding change in y is
Δy=(dy/ dx) Δx
In functional notation the corresponding
change in f is
Δf=f(x+Δx)-f(x) f ’(x) Δx
19

20. Approximation by Differentials Example 1
Suppose the total cost in dollars of
manufacturing q units of a certain
commodity is C(q)=3q2+5q+10. If the
current level of production is 40
units, estimate how the total cost will
change if 40.5 units are produced.
20

21. Approximation by Differentials Example 1
The current value of the variable is q=40
and the change in variable Δq=0.5. By the
approximation formula, the corresponding
change in cost is
ΔC=C(40.5)-C(40 C’(40) Δq
=C’(40) 0.5
Since C’(q)=6q+5 and
C’(40)=6 40+5=245
It follows that
ΔC C’(40) 0.5=245 0.5=$122.50
21

22. Approximation by Differentials Example 2
The daily output at a certain factory is
Q(L)=900L^(1/3) where L denotes the size
of the labor force measured in worker-
hours. Currently, 1,000 worker-hours of
labor are used each day. Use calculus to
estimate the number of additional worker-
hours of labor that will be needed to
increase daily output by 15 units.
(Answer: ΔL= 5 worker-hours)
22

23. Approximation of Percentage
change in
Percentage quantity
change
100
change size of
quantity
23

24. Approximation of Percentage
If Δx is a small change in x, the
corresponding percentage change in the
function f(x) is
change
f
% f 100
f x
f x x
100
f x
24

25. Example 2
Approximation of Percentage
The GNP of a certain country was
N(t)=t2+5t+200, billion dollars t years after
1990. Use calculus to estimate the
change
percentage change in the GNP during the
first quarter of 1998.
25

26. Example 3
Approximation of Percentage
N t t
% N  100
N t
change
N(t)=t2+5t+200; N’(t)=2t+5
with t=8, N(8)=82+5 8+200=304
N’(8)=2 8+5=21; Δt=0.25
Then
%ΔN 100(21)(0.25)/304=1.73 %
26

27. Approximation of Percentage
Example 4
At a certain factory, the daily output is
Q(K)=4,000K^(1/2) units, where K
denotes the Firm’s capital investment.
change
Use calculus to estimate the percentage
increase in output that will result from a
1 percent increase in capital investment.
(Answer: 0.5 %)
27

28. If C(x) is the total production cost
incurred by a manufacturer when x units
are produced then C’(x) is called the
Marginal cost
marginal cost.
If production is increased by 1 unit, then
Δx=1 and the approximation formula:
Δ C=C(x+ Δ x)-C(x) C’(x) Δ x
becomes
Δ C=C(x+1)-C(x) C’(x)
28

29. If R(x) is the total revenue derived from
sale of x units, then R’(x) is called the
marginal revenue.
Marginal cost
If sale is increased by 1 unit, then Δx 1
and the approximation formula:
ΔR R(x Δx) R(x) R’(x)Δx
becomes
ΔR R(x 1) R(x) R’(x)
29

30. Marginal Cost and Revenue
The marginal cost C’(x) is an
approximation to the cost
C(x 1) C(x) of producing the (x 1)st unit.
The marginal revenue R’(x) is an
approximation to the revenue
R(x 1) R(x) derived from the sale of the
(x 1)st unit.
30

31. Marginal Cost and Revenue Example 5
A manufacture estimates that when x units
of a particular commodity are
produced, the total cost will be
C(x) (1/8)x2 3x 98 dollars, and that
P(x) (1/3)(75 x) dollars per unit is the
price at which all x units will be sold.
a) Find the marginal cost and the
marginal revenue.
31

32. Marginal Cost and Revenue Example 5 (cont.)
b) Use marginal cost to estimate the
cost of producing the 9th unit.
c) What is the actual cost of producing
the 9th unit?
d) Use the marginal revenue to estimate
the revenue derived from the sale of
the 9th unit.
e) What is the actual revenue derived
from the sale of the 9th unit?
32

33. Marginal Cost and Revenue Example 5 (cont.)
a) The marginal cost is C’(x) x/4 3. Since
x units of the commodity are sold at a
price of
P(x) (75 x)/3 dollar per unit
the total revenue is
R(x) x P(x)=x(75 x)/3 25x x2/3
The marginal revenue is
R’(x) 25 2x/3
33

34. Marginal Cost and Revenue Example 5 (cont.)
b) The cost of producing the 9th unit is the
change in cost as x increase from 8 to
9 and can be estimated by the marginal
cost C’(8) 8/4 3 $5
c) The actual cost of producing the 9th unit
is ΔC C(9) C(8) $5.13
34

35. Marginal Cost and Revenue Example 5 (cont.)
d) The revenue obtained from the sale of
the 9th unit is approximated by the
marginal revenue:
R’(8) 25 (2/3)8 $19.67
e) The actual revenue obtained from the
sale of the 9th unit is
ΔR=R(9) R(8) $19.67
35

36. y y=f(x)
Differentials
P
D
Q
y
D dy
x x
x x+Dx
36

37. From approximation formula:
Δf f’(x) Δx or Δy f’(x) Δx
when Δx approaches zero, we can write
Differentials
dy y’dx, which is called differential of y.
37

38. Suppose y is a differentiable function of
u and u is a differentiable function of x.
Then y can be regarded as a function x
The Chain Rule
and
dy dy du
dx du dx
38

39. Example 1
Suppose that y u u and u x 3 17
Use the Chain Rule to find dy/dx and
The Chain Rule
evaluate it at x 2.
39

40. Example 2, 3
o Find dy/dx if y u/(u 1) and u 3x2 1
The Chain Rule
when x 1.
o Compute the derivatives of the following
functions
3
2 4
f x x 3x 2 f x 2x x
1
f x 5
2x 3
40

41. Example 4
An environmental study of a certain suburban
community suggests that the average daily
The Chain Rule
level of carbon monoxide in the air will be
C(p) (0.5p2 17) parts per million when the
population is p thousand. It is estimated that t
years from now, the population of the
community will be p(t) 3.1 0.1t2 thousand.
At what rate will the carbon monoxide level be
changing with respect to time 3 years from
now?
41

42. Example 4
The goal is to find dC/dt when t 3.
1
dC 1
The Chain Rule
2
0.5 p 17 2
0.5 2.p
dp 2
1
1 2
p 0.5 p 17 2
2
and
dp
0.2t
dt
42

43. Example 4
It follows from the chain rule that
1
dc 1
The Chain Rule
2
p 0.5 p 17 2
0.2t
dt 2
0.1pt
0.5 p 2 17
when t 3, p p(3) 3.1 0.1 32 4 . So,
dc 0.1 4 3 1.2
0.24
dt 0.5 4 2
17 25
43

44. The second derivative of a function is the
The Second Derivative
derivative of its derivative. If y f(x), the
second derivative is denoted by:
2
dy
2
or f x
dx
The second derivative gives the rate of
change of the rate of change of the
original function.
44

45. Example 1
The Second Derivative
Find both the first and second derivatives
of the functions:
f x x 3 12x 1
4 2
f x 5x 3x 3x 7
3x 2
f x 2
x 1
45

46. Example 2
An efficiency study of the morning shift at
The Second Derivative
a certain factory indicates that an average
worker who arrives on the job at 8:00AM.
Will have produced Q(t) t3 6t2 24t units
t hours later.
a) Compute the worker’s rate of
production at 11:00A.M
46

47. Example 2
b) At what rate is the worker’s rate of
The Second Derivative
production changing with respect to
time at 11:00A.M?
c) Use calculus to estimate the change in
the worker’s rate of production between
11:00 and 11:10A.M.
d) Compute the actual change in the
worker’s rate of production between
11:00 and 11:10A.M.
47

48. Example 2
a) The worker’s rate of production is the first
The Second Derivative
derivative
Q’(t) 3t2 12t 24
At 11:00 A.M.t 3 and the rate of
production is
Q’(3) 3 32 12 3 24 33 units per hour.
48

49. Example 2
The Second Derivative
b) The rate of change of the rate of
production is the second derivative
Q’’(t) 6t 12
At 11:00 A.M., the rate is
Q’’(3) 6 3 12 6 units /hour /hour.
49

50. Example 2
The Second Derivative
c) Note that 10 minutes is 1/6 hours, and
hence Δt 1/6 hour.
Change in rate of production is
ΔQ’ Q’’(t) Δt
ΔQ’ 6(1/6) 1 unit per hour.
50

51. Example 2
d) The actual change in the worker’s rate
The Second Derivative
of production between 11:00 and 11:10
A.M. is the difference between the
values of the rate Q’(t) when t 3 and
when t 19/6. That is
ΔQ’(t) Q’(19/6) Q’(3)
ΔQ’(t) 1.08 units per hour
51

52. Suppose that f is differentiable on the
interval (a,b).
The Concavity
a) If f is increasing on (a,b),then the graph
of f is concave upward on (a,b).
b) If f is decreasing on (a,b), then the
graph of f is concave downward on
(a,b).
52

53. Concave upward
The Concavity
(holds water)
Concave downward
(spills water)
53

54. A critical point of a function is a point on
its graph where either:
Critical Points
The derivative is zero, or
The derivative is undefined
The relative maxima and minima of the
function can occur only at critical points.
54

55. y
Concavity
Increasing, Concave upward x
f’ (x) >0, f ”(x)>0
55

56. Concavity
y
Concavity
x
Increasing, concave down
f x 0, f x 0
56

57. Concavity
y
Concavity
x
decreasing, concave up
f x 0, f x 0
57

58. Concavity
y
Concavity
x
decreasing, concave down
f x 0, f x 0
58

59. Second-Derivative Test
Suppose f’(a)=0.
If f’’(a)>0, then f has a relative minimum
at x=a.
If f’’(a)<0, then f has a relative
maximum at x=a.
However, if f’’(a)=0, the test is
inconclusive and f may have a relative
minimum, relative maximum, or no
relative extremum all at x=a.
59

60. Second-Derivative Test
y f a 0
f a 0
x
a
Relative maximum
60

61. Second-Derivative Test
y f a 0
f a 0
x
a
Relative minimum
61

62. Second-Derivative Test
y
f a 0
f a 0
x
a
Inflection Point
62

63. Second-Derivative Test
y
f a 0
f a 0
x
a
Inflection Point 63

64. Ex.2, page 33
Locate the local extrema for the
function
f(x)=(8/3)x3-x4
 Extrema:
the plural form of
extremum, the minimum or
maximum value of a function.
64

65. Ex.3, page 34
Use the second derivative test to
find the relative maxima and minima
of the function
f(x)=2x3+3x2-12x-7
65

66. Ex.4, page 34
Find the point of diminishing returns
for the sales function
S(x)=-0.02x3+3x2+100
where x represents thousands of
dollars spent on advertising, 0£x£80
and S is sales in thousands of dollars
for automobile tires.
66

67. Law of diminishing returns
Law of diminishing returns refers to a
situation in which a smaller result is
achieved for an increasing amount of
effort.
67

68. Average cost per unit (AC)
Average cost per unit (AC) is the total
cost divided by the number of units
produced. Hence, if C(q) denotes the
total cost of producing q units of item, the
average cost per unit is
C q
AC q
q
68

69. Average Cost and Marginal
Cost
Suppose AC and MC denote the average
cost and marginal cost respectively. Then
 AC is decreasing when MC<AC
 AC is increasing when MC>AC
 AC has (first-order) critical point (usually
relative minimum) when MC=AC
69

70. Elasticity of Demand
p: the price,
q: the corresponding number of units
demand
Δp: a small chance in price,
then the percentage change in q is
dq dp p
% q  100
q
70

71. Elasticity of Demand
If the change in p is a 1-percent
increase, then Δp=0.01p and
dq dp 0.01p
% q  100
q
p dq
q dp
71

72. Elasticity of Demand
If q denotes the demand for a commodity
and p its price, the elasticity of demand, is
given by
=(p/q)(dq/dp)
It is the percentage change in demand due
to a 1 percentage increase in price.
72

73. Ex. 1, page 35
Suppose the demand q and price p for a
certain commodity are related by the linear
equation q=240-2p (for 0£p£120 ).
a) Express the elasticity of demand as a
function of p.
b) Calculate the elasticity of demand
when the price is p=100. Interpret the
answer.
73

74. Ex.1, page 35 (Continued)
c) Calculate the elasticity of demand when
the price is p=50. Interpret the answer.
d) At what price is the elasticity of demand
equal to -1?
74

75. Ex.1, page 35 (Continued)
a) The elasticity of demand is
p dq p
2
q dp q
2p
240 2p
p
120 p
75

76. Ex.1, page 35 (Continued)
b) When p=100, the elasticity of demand is
p
120 p
100
5
120 100
76

77. Ex.1, page 35 (Continued)
That is, when the price is p=100, a 1-
percent increase in price will produce a
decrease in demand of approximately 5
percent.
77

78. Ex.1, page 35 (Continued)
c) When p=50, the elasticity of demand is
p
120 p
50
120 50
0.71
78

79. Levels of Elasticity of Demand
In general, the elasticity of demand is
negative, since demand decreases as
price increases.
79

80. Levels of Elasticity of Demand
(Continued)
If | |>1, demand is said to be elastic
with respect to price.
If | |<1, demand is said to be inelastic
with respect to price.
If | |=1, demand is said to be unit
elasticity with respect to price.
80

81. Elasticity and the Total Revenue
If R denotes the total revenue, p the price
per unit, and q the number of units sold
(i.e. the demand), then we can obtain.
If demand is inelastic (| |<1 ), total
revenue increases as price increases.
If demand is elastic (| |>1), total revenue
decreases as price increases.
81

82. Ex. 2, page 36
Suppose the demand q and price p for a
certain commodity are related by the
equation q=300-p2, 0 p (300)
a) Determine where the demand is
elastic, inelastic, and of unit elasticity
with respect to price.
82

83. Ex. 2, page 36 (cont.)
b) Use the results of part a) to
describe the behavior of the total
revenue as a function of price.
c) Find the total revenue function
explicitly and use its first derivative
to determine its intervals of
increase and decrease and the
price at which revenue is
maximized.
83

84. Ex. 2, page 36 (cont.)
a) The elasticity of demand is
p dq
q dp
p
2
2p
300 p
2
2p
2
300 p
84

85. Ex. 2, page 36 (cont.)
The demand is of unit elasticity when
|h|=1, that is, when
2
2p
2
1
300 p
2
p 100
p 10
of which only p=10 is in the relevant
interval 0£p£Ö300.
85

86. Ex. 2, page 36 (cont.)
If 0£p<10,
2 2
2p 2 10
2 2
1
300 p 300 10
and hence the demand is inelastic.
86

87. Ex. 2, page 36 (cont.)
If 10<p<Ö300,
2 2
2p 2 10
2 2
1
300 p 300 10
and hence the demand is elastic.
87

88. Ex. 2, page 36 (cont.)
b) The total revenue is an increasing
function of p when demand is
inelastic, that is, on the interval
0£p<10 and a decreasing function
of p when demand is elastic, that
is, on the interval 10<p<Ö300.
At the price p=10 of unit elasticity,
the revenue function has a relative
maximum.
88

89. Ex. 2, page 36 (cont.)
c) The revenue function is R=pq or
2 3
R p p 300 p 300p p
Its derivative is
2
R p 300 3p
3 10 p 10 p
which is zero when p=± 10, of
which only p=10 is in the relevant
interval .
89

90. Ex. 2, page 36 (cont.)
o On the interval 0£p<10, R’(p)>0, so
R(p) is increasing.
o On the interval 10<p£Ö300, R’(p)<0,
so R(p) is decreasing.
o At the critical value p=10, R(p) stops
increasing and starts decreasing and
hence has a relative maximum.
90

91. Thank you for your attention!
THE END!
91