Lesson 5.
Differentiation Rules for
Exponential and
Logarithmic Functions
Basic Calculus
Capstone Project
Science, Technology, Engineering, and Mathematics
To know the age of
carbon-based
materials,
archeologists use
carbon-dating, and
they use
mathematical models
for this.
2
These models apply
differentiation to
investigate the rate at
which the carbon-
based substances
changes relative to the
quantity of the
substance present at
that time.
3
In this lesson, we will learn how to derive differentiation
rules for exponential and logarithmic functions and apply
these rules to differentiate such functions.
4
How will you obtain the
derivative of exponential and
logarithmic functions?
5
Learning Competency
At the end of the lesson, you should be able to do the following:
Apply the differentiation rules in computing
the derivative of exponential and logarithmic
functions (STEM_BC11D-IIIf-3).
6
Learning Objectives
At the end of the lesson, you should be able to do the following:
● Derive the differentiation rules for the derivative
of exponential and logarithmic functions.
● Apply the differentiation rules to compute for the
derivative of exponential and logarithmic
functions.
7
Derivative of Natural Exponential Functions
The natural exponential function is an exponential
function whose base is the natural number 𝑒 where
𝑒 ≈ 2.718281 … It is written in the form 𝑓 𝑥 = 𝑒 𝑥 .Its
derivative is given by
𝒅 𝒙
𝒆 = 𝒆𝒙 .
𝒅𝒙
8
Derivative of Natural Exponential Functions
Example:
Find the derivative of the function 𝑓 𝑥 = 4𝑒 𝑥 .
9
Derivative of Natural Exponential Functions
Example:
Find the derivative of the function 𝑓 𝑥 = 4𝑒 𝑥 .
𝒇′ 𝒙 = 𝟒𝒆𝒙
10
Derivative of a General Exponential Functions
An exponential function is a function of the form
𝑓 𝑥 = 𝑎 𝑥 where 𝑎 is the base. In this function, 𝑎 > 0
and 𝑎 ≠ 1. Its derivative is given by
𝒅 𝒙
𝒂 = 𝒂𝒙 𝐥𝐧 𝒂 .
𝒅𝒙
11
Derivative of a General Exponential Functions
Example:
Find the derivative of the function 𝑓 𝑥 = 3𝑥 .
12
Derivative of a General Exponential Functions
Example:
Find the derivative of the function 𝑓 𝑥 = 3𝑥 .
𝒇′ 𝒙 = 𝟑𝒙 𝐥𝐧 𝟑
13
How will you obtain the
derivative of the function
𝒇 𝒙 = −𝟒𝒆𝒙 + 𝟐𝒙 ?
14
Let’s Practice!
What is the derivative of the function 𝒇 𝒙 = 𝟓𝒙 𝒆𝒙 ?
15
Let’s Practice!
What is the derivative of the function 𝒇 𝒙 = 𝟓𝒙 𝒆𝒙 ?
𝒇′ 𝒙 = 𝟓𝒙 𝒆𝒙 𝟏 + 𝐥𝐧 𝟓
16
Try It!
What is the derivative of the function
𝒇 𝒙 = 𝟏𝟐𝒙 𝒆𝒙 ?
17
Let’s Practice!
𝒆𝒙 −𝟏
Find 𝒚′ given that 𝒚 = .
𝒆𝒙 +𝟏
18
Let’s Practice!
𝒆𝒙 −𝟏
Find 𝒚′ given that 𝒚 = .
𝒆𝒙 +𝟏
𝟐𝒆𝒙
𝒚′ = 𝒙 𝟐
𝒆 +𝟏
19
Try It!
𝒆𝒙 +𝟐
Find 𝒚′ given that 𝒚 = 𝒙 .
𝒆 −𝟐
20
Let’s Practice!
𝒆𝒙 𝒙−𝟏
Determine 𝒈′ 𝒙 provided that 𝒈 𝒙 = .
𝒆𝒙 +𝟐
21
Let’s Practice!
𝒆𝒙 𝒙−𝟏
Determine 𝒈′ 𝒙 provided that 𝒈 𝒙 = .
𝒆𝒙 +𝟐
𝒙 𝟐𝒙
𝟐𝒙𝒆 + 𝒆
𝒈′ 𝒙 =
𝒆𝒙 + 𝟐 𝟐
22
Try It!
Determine 𝒈′ 𝒙 provided that
𝒆𝒙 𝒙+𝟑
𝒈 𝒙 = .
𝒆𝒙 −𝟑
23
Derivative of a Natural Logarithmic Function
A natural logarithmic function is a logarithmic
function whose base is the natural number 𝑒. Thus,
the function is of the form 𝑓 𝑥 = log 𝑒 𝑥 or 𝑓 𝑥
= ln 𝑥. Its derivative is given by
𝒅 𝟏
𝐥𝐧 𝒙 = .
𝒅𝒙 𝒙
24
Derivative of a Natural Logarithmic Function
Example:
Find the derivative of the function 𝑓 𝑥 = 3 ln 𝑥.
25
Derivative of a Natural Logarithmic Function
Example:
Find the derivative of the function 𝑓 𝑥 = 3 ln 𝑥.
𝟑
𝒇′ 𝒙 =
𝒙
26
Derivative of a General Logarithmic Function
A logarithmic function is a function of the form
𝑓 𝑥 = log 𝑎 𝑥 , where 𝑎 > 0 , 𝑎 ≠ 1 , and 𝑥 is any
positive real number. Its derivative is given by
𝒅 𝟏
𝐥𝐨𝐠 𝒂 𝒙 = .
𝒅𝒙 𝒙 𝐥𝐧 𝒂
27
Derivative of a General Logarithmic Function
Example:
Find the derivative of the function 𝑓 𝑥 = 5 log 3 𝑥.
28
Derivative of a General Logarithmic Function
Example:
Find the derivative of the function 𝑓 𝑥 = 5 log 3 𝑥.
𝟓
𝒇′ 𝒙 =
𝒙 𝐥𝐧 𝟑
29
Let’s Practice!
What is the derivative of the function 𝒇 𝒙 = 𝟒𝒙𝟐 𝐥𝐧 𝒙?
30
Let’s Practice!
What is the derivative of the function 𝒇 𝒙 = 𝟒𝒙𝟐 𝐥𝐧 𝒙?
𝒇′ 𝒙 = 𝟒𝒙 + 𝟖𝒙 𝐥𝐧 𝒙
31
Try It!
What is the derivative of the function
𝒇 𝒙 = 𝟓𝒙𝟑 𝐥𝐧 𝒙?
32
Let’s Practice!
𝐥𝐧 𝒙
Find 𝒚′ given that 𝒚 = .
𝒆𝒙
33
Let’s Practice!
𝐥𝐧 𝒙
Find 𝒚′ given that 𝒚 = .
𝒆𝒙
𝟏 − 𝒙 𝐥𝐧 𝒙
𝒚′ =
𝒙𝒆𝒙
34
Try It!
𝟑 𝐥𝐧 𝒙
Find 𝒚′ given that 𝒚 = .
𝟐𝒆𝒙
35
Let’s Practice!
𝒙 𝐥𝐨𝐠 𝟓 𝒙
Determine 𝒈′ 𝒙 provided that 𝒈 𝒙 = .
𝒆𝒙 +𝟏
36
Let’s Practice!
𝒙 𝐥𝐨𝐠 𝟓 𝒙
Determine 𝒈′ 𝒙 provided that 𝒈 𝒙 = .
𝒆𝒙 +𝟏
𝟏 + 𝐥𝐧 𝒙 − 𝒙 𝐥𝐧 𝒙
𝒈′ 𝒙 =
𝒆𝒙 𝐥𝐧 𝟓
37
Try It!
Determine 𝒈′ 𝒙 provided that
𝑥 2 log3 𝑥
𝒈 𝒙 = .
𝑒𝑥
38
Remember
You can always use some properties
and rules of logarithms to simplify your
answers.
● Product Rule
log 𝑎 𝑀 + log 𝑎 𝑁 = log 𝑎 𝑀𝑁
39
Remember
● Quotient Rule
𝑀
log 𝑎 𝑀 − log 𝑎 𝑁 = log 𝑎
𝑁
● Power Rule
𝑥
𝑥 log 𝑎 𝑀 = log 𝑎 𝑀
40
Remember
● Change-of-Base Formula
log 𝑥 ln 𝑥
log 𝑎 𝑥 = =
log 𝑎 ln 𝑎
41
Let’s Practice!
𝒆𝒙
Find 𝒇′ 𝟏 given that 𝒇 𝒙 = .
𝟑𝒙
42
Let’s Practice!
𝒆𝒙
Find 𝒇′ 𝟏 given that 𝒇 𝒙 = .
𝟑𝒙
𝒆−𝒆 𝐥𝐧 𝟑
𝒇′ 𝟏 =
𝟑
43
Try It!
𝒆𝒙
Find 𝒇′ 𝟏 given that 𝒇 𝒙 = 𝒙 .
𝟐
44
Let’s Practice!
Find the equation of the line tangent to the curve
𝒇 𝒙 = 𝐥𝐨𝐠 𝟐 𝒙 at 𝒙 = 𝟖.
45
Let’s Practice!
Find the equation of the line tangent to the curve
𝒇 𝒙 = 𝐥𝐨𝐠 𝟐 𝒙 at 𝒙 = 𝟖.
𝑥−8+24 ln 2
𝑦=
8 ln 2
46
Try It!
Find the equation of the line tangent to
the curve 𝒇 𝒙 = 𝐥𝐨𝐠 𝟑 𝒙 at 𝒙 = 𝟗.
47
Lesson 5.4
Differentiation Rules for
Trigonometric Functions
Basic Calculus
Capstone Project
Science, Technology, Engineering, and Mathematics
Waves are significant
in human lives. We
can see because of
visible light, which is
an example of a wave.
49
Mobile phones,
televisions, and radios
would not work
without microwaves
and radio waves,
which is usually
coming from a signal
tower.
50
These waves can be mathematically represented on a
coordinate plane using trigonometric functions like the sine
function.
51
In this lesson, we will learn how to derive the differentiation
rules for trigonometric functions and how these rules are
applied to differentiate functions involving trigonometric
functions.
52
How will you solve for the
derivative of trigonometric
functions?
53
Learning Competency
At the end of the lesson, you should be able to do the following:
Apply the differentiation rules in computing
the derivative of trigonometric functions
(STEM_BC11D-IIIf-3)
54
Learning Objectives
At the end of the lesson, you should be able to do the following:
● Derive the differentiation rules for trigonometric
functions.
● Apply the differentiation rules in computing the
derivative of trigonometric functions.
55
Derivative of Trigonometric Functions
The differentiation rules for some of the trigonometric
functions can also be derived using the limit definition of
derivative.
𝑓 𝑥+ℎ −𝑓 𝑥
𝑓′ 𝑥 = lim
ℎ→0 ℎ
56
Derivative of Trigonometric Functions
Special limits will come in handy when some of the
differentiation rules of the six trigonometric functions are
derived.
sin 𝑥
lim =1
𝑥→0 𝑥
1 − cos 𝑥
lim =0
𝑥→0 𝑥
57
Remember
We can use trigonometric identities to
derive the derivative rules for some of
the trigonometric functions. We can
also use these identities to simplify the
derivatives of trigonometric functions.
58
Remember
● Reciprocal Identities
1 1
sin 𝑥 = cot 𝑥 =
csc 𝑥 tan 𝑥
1 1
cos 𝑥 = sec 𝑥 =
sec 𝑥 cos 𝑥
1 1
tan 𝑥 = csc 𝑥 =
cot 𝑥 sin 𝑥
59
Remember
● Quotient Identities
sin 𝑥 cos 𝑥
tan 𝑥 = cot 𝑥 =
cos 𝑥 sin 𝑥
60
Remember
● Pythagorean Identities
sin2 𝑥 + cos 2 𝑥 = 1
Other forms:
sin2 𝑥 = 1 − cos 2 𝑥
cos 2 𝑥 = 1 − sin2 𝑥
61
Remember
● Pythagorean Identities
1 + tan2 𝑥 = sec 2 𝑥
Other form:
tan2 𝑥 = sec 2 𝑥 − 1
62
Remember
● Pythagorean Identities
1 + cot 2 𝑥 = csc 2 𝑥
Other form:
cot 2 𝑥 = csc 2 𝑥 − 1
63
Remember
● Sum or Difference Identities
sin 𝑥 ± 𝑦 = sin 𝑥 cos 𝑦 ± cos 𝑥 sin 𝑦
cos 𝑥 ± 𝑦 = cos 𝑥 cos 𝑦 ∓ sin 𝑥 sin 𝑦
tan 𝑥 ± tan 𝑦
tan 𝑥 ± 𝑦 =
1 ∓ tan 𝑥 tan 𝑦
64
Derivative of a Sine Function
The derivative of the sine function 𝑓 𝑥 = sin 𝑥 is given by
𝒅
𝐬𝐢𝐧 𝒙 = 𝐜𝐨𝐬 𝒙.
𝒅𝒙
65
Derivative of a Sine Function
Example:
Find the derivative of 𝑓 𝑥 = 4 sin 𝑥.
66
Derivative of a Sine Function
Example:
Find the derivative of 𝑓 𝑥 = 4 sin 𝑥.
𝒇′ 𝒙 = 𝟒 𝐜𝐨𝐬 𝒙
67
Derivative of a Cosine Function
The derivative of the cosine function 𝑓 𝑥 = cos 𝑥 is given
by
𝒅
𝐜𝐨𝐬 𝒙 = − 𝐬𝐢𝐧 𝒙.
𝒅𝒙
68
Derivative of a Cosine Function
Example:
Find the derivative of 𝑓 𝑥 = 10 cos 𝑥.
69
Derivative of a Cosine Function
Example:
Find the derivative of 𝑓 𝑥 = 10 cos 𝑥.
𝒇′ 𝒙 = −𝟏𝟎 𝐬𝐢𝐧 𝒙
70
Derivative of a Tangent Function
The derivative of the tangent function 𝑓 𝑥 = tan 𝑥 is given
by
𝒅
𝐭𝐚𝐧 𝒙 = 𝐬𝐞𝐜 𝟐 𝒙.
𝒅𝒙
71
Derivative of a Tangent Function
Example:
Find the derivative of 𝑓 𝑥 = 3 tan 𝑥 + 4.
72
Derivative of a Tangent Function
Example:
Find the derivative of 𝑓 𝑥 = 3 tan 𝑥 + 4.
𝒇′ 𝒙 = 𝟑 𝐬𝐞𝐜 𝟐 𝒙
73
Derivative of a Cotangent Function
The derivative of the cotangent function 𝑓 𝑥 = cot 𝑥 is
given by
𝒅
𝐜𝐨𝐭 𝒙 = − 𝐜𝐬𝐜 𝟐 𝒙.
𝒅𝒙
74
Derivative of a Cotangent Function
Example:
Find the derivative of 𝑓 𝑥 = −6 cot 𝑥 − 12.
75
Derivative of a Cotangent Function
Example:
Find the derivative of 𝑓 𝑥 = −6 cot 𝑥 − 12.
𝒇′ 𝒙 = 𝟔 𝐜𝐬𝐜 𝟐 𝒙
76
Derivative of a Secant Function
The derivative of the secant function 𝑓 𝑥 = sec 𝑥 is given
by
𝒅
𝐬𝐞𝐜 𝒙 = 𝐬𝐞𝐜 𝒙 𝐭𝐚𝐧 𝒙.
𝒅𝒙
77
Derivative of a Secant Function
Example:
Find the derivative of 𝑓 𝑥 = 2 sec 𝑥.
78
Derivative of a Secant Function
Example:
Find the derivative of 𝑓 𝑥 = 2 sec 𝑥.
𝒇′ 𝒙 = 𝟐 𝐬𝐞𝐜 𝒙 𝐭𝐚𝐧 𝒙
79
Derivative of a Cosecant Function
The derivative of the cosecant function 𝑓 𝑥 = csc 𝑥 is
given by
𝒅
𝐜𝐬𝐜 𝒙 = − 𝐜𝐬𝐜 𝒙 𝐜𝐨𝐭 𝒙.
𝒅𝒙
80
Derivative of a Cosecant Function
Example:
Find the derivative of 𝑓 𝑥 = 3 csc 𝑥.
81
Derivative of a Cosecant Function
Example:
Find the derivative of 𝑓 𝑥 = 3 csc 𝑥.
𝒇′ 𝒙 = −𝟑 𝐜𝐬𝐜 𝒙 𝐜𝐨𝐭 𝒙
82
How will you find the derivative
of the function
𝒇 𝒙 = 𝟏𝟓 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒙 + 𝟒 𝐭𝐚𝐧 𝒙 ?
83
Let’s Practice!
What is the derivative of the function
𝒇 𝒙 = 𝟏𝟐 𝐬𝐢𝐧 𝒙 − 𝟐𝟔 𝐜𝐨𝐬 𝒙?
84
Let’s Practice!
What is the derivative of the function
𝒇 𝒙 = 𝟏𝟐 𝐬𝐢𝐧 𝒙 − 𝟐𝟔 𝐜𝐨𝐬 𝒙?
𝒇′ 𝒙 = 𝟏𝟐 𝐜𝐨𝐬 𝒙 + 𝟐𝟔 𝐬𝐢𝐧 𝒙
85
Try It!
What is the derivative of the function
𝒇 𝒙 = 𝟓 𝐜𝐨𝐬 𝒙 + 𝟑 𝐬𝐢𝐧 𝒙?
86
Let’s Practice!
Find the derivative of the function 𝒇 𝒙 = 𝟒𝒙𝟐 𝐭𝐚𝐧 𝒙.
87
Let’s Practice!
Find the derivative of the function 𝒇 𝒙 = 𝟒𝒙𝟐 𝐭𝐚𝐧 𝒙.
𝒇′ 𝒙 = 𝟒𝒙𝟐 𝐬𝐞𝐜 𝟐 𝒙 + 𝟖𝒙 𝐭𝐚𝐧 𝒙
88
Try It!
Find the derivative of the function
𝒇 𝒙 = −𝟐𝒙𝟐 𝐜𝐨𝐭 𝒙.
89
Let’s Practice!
Determine 𝒈′ 𝒙 if 𝒈 𝒙 = 𝟑 𝐬𝐞𝐜 𝒙 𝐭𝐚𝐧 𝒙. Express your
answer using one trigonometric function.
90
Let’s Practice!
Determine 𝒈′ 𝒙 if 𝒈 𝒙 = 𝟑 𝐬𝐞𝐜 𝒙 𝐭𝐚𝐧 𝒙 .Express your
answer using one trigonometric function.
𝒈′ 𝒙 = 𝟔 𝐬𝐞𝐜 𝟑 𝒙 − 𝟑 𝐬𝐞𝐜 𝒙
91
Try It!
Determine 𝒈′ 𝒙 if 𝒈 𝒙 = −𝟔 𝐜𝐬𝐜 𝒙 𝐜𝐨𝐭 𝒙.
Express your answer using one
trigonometric function.
92
Let’s Practice!
𝟐 𝐬𝐢𝐧 𝒙
What is 𝒚′ given that 𝒚 = ?
𝐜𝐨𝐬 𝒙+𝟏
93
Let’s Practice!
𝟐 𝐬𝐢𝐧 𝒙
What is 𝒚′ given that 𝒚 = ?
𝐜𝐨𝐬 𝒙+𝟏
𝟐
𝒚′ =
𝐜𝐨𝐬 𝒙 + 𝟏
94
Try It!
𝟐 𝐜𝐨𝐬 𝒙
What is 𝒚′ given that 𝒚 = ?
𝐬𝐢𝐧 𝒙+𝟏
95
Let’s Sum It Up!
The following formulas are the derivatives of exponential
and logarithmic functions derived from the limit definition
of a derivative:
● Derivative of a Natural Exponential Function
𝑑 𝑥
𝑒 = 𝑒𝑥
𝑑𝑥
● Derivative of a General Exponential Function
𝑑 𝑥
𝑎 = 𝑎 𝑥 ln 𝑎
𝑑𝑥 96
Let’s Sum It Up!
● Derivative of a Natural Logarithmic Function
𝑑 1
ln 𝑥 =
𝑑𝑥 𝑥
● Derivative of a General Logarithmic Function
𝑑 1
log 𝑎 𝑥 =
𝑑𝑥 𝑥 ln 𝑎
97
Key Formulas
Concept Formula Description
Use this formula to
Derivative of a 𝑑 𝑥
𝑒 = 𝑒𝑥 solve for the derivative
Natural Exponential 𝑑𝑥
of a natural
Function
where 𝑒 is the natural exponential function.
number (𝑒 ≈ 2.818281 …)
Use this formula to
Derivative of a 𝑑 𝑥
𝑎 = 𝑎 𝑥 ln 𝑎 solve for the derivative
General Exponential 𝑑𝑥
of a general
Function
where 𝑎 > 0 and 𝑎 ≠ 1 exponential function.
98
Key Formulas
Concept Formula Description
Use this formula to
Derivative of a 𝑑 1
ln 𝑥 = solve for the derivative
Natural Logarithmic 𝑑𝑥 𝑥
of a natural logarithmic
Function
where 𝑥 > 0 function.
Use this formula to
Derivative of a 𝑑 1
log 𝑎 𝑥 = solve for the derivative
General Logarithmic 𝑑𝑥 𝑥 ln 𝑎
of a general
Function
where 𝑎 > 0, 𝑎 ≠ 1, and logarithmic function.
𝑥>0
99
Let’s Sum It Up!
● Derivative of a Sine Function
𝑑
sin 𝑥 = cos 𝑥
𝑑𝑥
100
Let’s Sum It Up!
● Derivative of a Cosine Function
𝑑
cos 𝑥 = − sin 𝑥
𝑑𝑥
101
Let’s Sum It Up!
● Derivative of a Tangent Function
𝑑
tan 𝑥 = sec 2 𝑥
𝑑𝑥
102
Let’s Sum It Up!
● Derivative of a Cotangent Function
𝑑
cot 𝑥 = − csc 2 𝑥
𝑑𝑥
103
Let’s Sum It Up!
● Derivative of a Secant Function
𝑑
sec 𝑥 = sec 𝑥 tan 𝑥
𝑑𝑥
104
Let’s Sum It Up!
● Derivative of a Cosecant Function
𝑑
csc 𝑥 = − csc 𝑥 cot 𝑥
𝑑𝑥
105
Key Formulas
Concept Formula Description
Use this formula to solve
Derivative of a Sine 𝑑 for the derivative of a
Function sin 𝑥 = cos 𝑥
𝑑𝑥 function that involves a
sine function.
Use this formula to solve
Derivative of a 𝑑 for the derivative of a
Cosine Function cos 𝑥 = − sin 𝑥
𝑑𝑥 function that involves a
cosine function.
106
Key Formulas
Concept Formula Description
Use this formula to solve
Derivative of a 𝑑 for the derivative of a
Tangent Function tan 𝑥 = sec 2 𝑥
𝑑𝑥 function that involves a
tangent function.
Use this formula to solve
Derivative of a 𝑑 for the derivative of a
Cotangent Function cot 𝑥 = − csc 2 𝑥
𝑑𝑥 function that involves a
cotangent function.
107
Key Formulas
Concept Formula Description
Use this formula to solve
Derivative of a 𝑑
sec 𝑥 = sec 𝑥 tan 𝑥 for the derivative of a
Secant Function 𝑑𝑥 function that involves a
secant function.
Use this formula to solve
Derivative of a 𝑑
csc 𝑥 = − csc 𝑥 cot 𝑥 for the derivative of a
Cosecant Function 𝑑𝑥
function that involves a
cosecant function.
108
Photo Credit Bibliography
Edwards, C.H., and David E. Penney. Calculus: Early
Transcendentals. 7th ed. Upper Saddle River, New Jersey:
Pearson/Prentice Hall, 2008.
Larson, Ron H., and Bruce H. Edwards. Essential Calculus: Early
● Slide 2-3: Seymouria Fossil.jpg by Sanjay Acharya is Transcendental Functions. Boston: Houghton Mifflin, 2008.
licensed under CC BY-SA 4.0 via Wikimedia Commons.
Leithold, Louis. The Calculus 7. New York: HarperCollins
College Publ., 1997.
Smith, Robert T., and Roland B. Milton. Calculus. New York:
McGraw Hill, 2012.
Tan, Soo T. Applied Calculus for the Managerial, Life, and Social
Sciences: A Brief Approach. Australia: Brooks/Cole Cengage
Learning, 2012.
109
Bibliography
Edwards, C.H., and David E. Penney. Calculus: Early Transcendentals. 7th ed. Upper Saddle River, New
Jersey: Pearson/Prentice Hall, 2008.
Larson, Ron H., and Bruce H. Edwards. Essential Calculus: Early Transcendental Functions. Boston:
Houghton Mifflin, 2008.
Leithold, Louis. The Calculus 7. New York: HarperCollins College Publ., 1997.
Smith, Robert T., and Roland B. Milton. Calculus. New York: McGraw Hill, 2012.
Tan, Soo T. Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach. Australia:
Brooks/Cole Cengage Learning, 2012.
110
