Basic Calculus Basic Differentiation Rules

1. Capstone Project
Science, Technology, Engineering, and Mathematics
Basic Calculus
Science, Technology, Engineering, and Mathematics
Lesson 5.1
The Basic Differentiation
Rules

2. 2
A car speedometer
tells us our current
speed when we drive.
However, this speed is
not either our
constant speed nor
our average speed
when we drive.

3. 3
This speed is called the instantaneous rate of
change, or simply the instantaneous speed.
This concept is related to limits and
derivatives.

4. 4
In this lesson, we are going to derive basic
differentiation rules with the aid of the limit
definition of derivative to enable us to solve for the
derivatives of algebraic functions.

5. 5
How do we differentiate
functions without the use of the
limit definition of derivative?

6. Learning Competencies
At the end of the lesson, you should be able to do the following:
6
● Derive the differentiation rules (STEM_BC11D-
IIIf-2).
● Apply the differentiation rules in computing
the derivative of an algebraic, exponential,
and trigonometric functions (STEM_BC11D-IIIf-
3).

7. Learning Objectives
At the end of the lesson, you should be able to do the following:
7
● Derive the basic differentiation rules (Constant
Rule, Power Rule, Constant Multiple Rule, Sum or
Difference Rule).
● Apply the basic differentiation rules in solving for
the derivatives of functions.

8. 8
Constant Rule
The derivative of a constant function is 0. If 𝑐 is a constant,
then
𝒅
𝒅𝒙
𝒄 = 𝟎.
The Basic Differentiation Rules

9. 9
Proof:
Let 𝑓 𝑥 = 𝑐. We use the limit definition of a derivative to
derive the Constant Rule.
The Basic Differentiation Rules

10. 10
Proof:
The Basic Differentiation Rules
𝑑
𝑑𝑥
𝑐 = lim
ℎ→0
𝑓 𝑥 + ℎ − 𝑓 𝑥
ℎ
Limit Definition of
Derivative

11. 11
Proof:
The Basic Differentiation Rules
𝑑
𝑑𝑥
𝑐 = lim
ℎ→0
𝑓 𝑥 + ℎ − 𝑓 𝑥
ℎ
= lim
ℎ→0
𝑐 − 𝑐
ℎ
= lim
ℎ→0
0
Limit Definition of
Derivative
Subtraction

12. 12
Proof:
The Basic Differentiation Rules
𝑑
𝑑𝑥
𝑐 = lim
ℎ→0
𝑓 𝑥 + ℎ − 𝑓 𝑥
ℎ
= lim
ℎ→0
𝑐 − 𝑐
ℎ
= lim
ℎ→0
0
= 0
Limit Definition of
Derivative
Subtraction
Constant Law of Limits

13. 13
Proof:
Therefore,
𝑑
𝑑𝑥
𝑐 = 0.
The Basic Differentiation Rules

14. 14
Example:
a. If 𝑓 𝑥 = 6, then 𝑓′ 𝑥 = 0.
b. Given than 𝑦 = 24, 𝑦′ = 0.
The Basic Differentiation Rules

15. 15
Power Rule
If 𝑘 is a real number, then the derivative of the function
𝑓 𝑥 = 𝑥𝑘
is
𝒅
𝒅𝒙
𝒙𝒌
= 𝒌𝒙𝒌−𝟏
.
The Basic Differentiation Rules

16. 16
Proof:
We will prove the Power Rule for positive integer
exponent. The proof for negative integer and real number
exponents need more advanced techniques (Quotient
Rule and implicit differentiation, respectively) which will be
discussed in the succeeding lessons on derivatives.
The Basic Differentiation Rules

17. 17
Proof:
Let 𝑓 𝑥 = 𝑥𝑘, where 𝑘 is any positive integer. We use the
limit definition of a derivative to derive the Power Rule.
The Basic Differentiation Rules

18. 18
The Basic Differentiation Rules
𝑑
𝑑𝑥
𝑥𝑘 = lim
ℎ→0
𝑥 + ℎ 𝑘
− 𝑥𝑘
ℎ
Limit Definition of
Derivative

19. 19
Proof:
The Basic Differentiation Rules
𝑑
𝑑𝑥
𝑥𝑘
= lim
ℎ→0
𝑥𝑘
+ 𝑘𝑥𝑘−1
ℎ +
𝑘 𝑘 − 1
2
𝑥𝑘−2
ℎ2
+ ⋯ + 𝑘𝑥ℎ𝑘−1
+ ℎ𝑘
− 𝑥𝑘
ℎ
Binomial Expansion

20. 20
Proof:
The Basic Differentiation Rules
𝑑
𝑑𝑥
𝑥𝑘 = lim
ℎ→0
𝑘𝑥𝑘−1ℎ +
𝑘 𝑘 − 1
2
𝑥𝑘−2ℎ2 + ⋯ + 𝑘𝑥ℎ𝑘−1 + ℎ𝑘
ℎ
Subtraction

21. 21
Proof:
The Basic Differentiation Rules
𝑑
𝑑𝑥
𝑥𝑘 = lim
ℎ→0
ℎ 𝑘𝑥𝑘−1 +
𝑘 𝑘 − 1
2
𝑥𝑘−2ℎ + ⋯ + 𝑘𝑥ℎ𝑘−2 + ℎ𝑘−1
ℎ
Factor and cancel ℎ.

22. 22
Proof:
The Basic Differentiation Rules
𝑑
𝑑𝑥
𝑥𝑘
= lim
ℎ→0
𝑘𝑥𝑘−1
+
𝑘 𝑘 − 1 𝑥𝑘−2
2
𝑥𝑘−2
ℎ + ⋯ + 𝑘𝑥ℎ𝑘−2
+ ℎ𝑘−1
= 𝑘𝑥𝑘−1
Evaluation of limit

23. 23
Proof:
Therefore,
𝑑
𝑑𝑥
𝑥𝑘 = 𝑘𝑥𝑘−1.
The Basic Differentiation Rules

24. 24
Example:
Find the derivative of 𝑓 𝑥 = 𝑥4.
The Basic Differentiation Rules

25. 25
Example:
Find the derivative of 𝑓 𝑥 = 𝑥4.
The derivative of 𝑓 𝑥 = 𝑥4
is 𝑓′
𝑥 = 4𝑥3
.
The Basic Differentiation Rules

26. 26
Constant Multiple Rule
If 𝑓 is differentiable and 𝑐 is a constant, then the derivative
of the function 𝑐 ∙ 𝑓 is
𝒅
𝒅𝒙
𝒄𝒇 𝒙 = 𝒄𝒇′ 𝒙 .
The Basic Differentiation Rules

27. 27
Proof:
The Basic Differentiation Rules
𝑑
𝑑𝑥
𝑐𝑓 𝑥 = lim
ℎ→0
𝑐𝑓 𝑥 + ℎ − 𝑐𝑓 𝑥
ℎ
Limit Definition of Derivative

28. 28
Proof:
The Basic Differentiation Rules
𝑑
𝑑𝑥
𝑐𝑓 𝑥 = lim
ℎ→0
𝑐
𝑓 𝑥 + ℎ − 𝑓 𝑥
ℎ
Factor 𝑐.

29. 29
Proof:
The Basic Differentiation Rules
𝑑
𝑑𝑥
𝑐𝑓 𝑥 = 𝑐 lim
ℎ→0
𝑓 𝑥 + ℎ − 𝑓 𝑥
ℎ
Constant Multiple Law for Limits

30. 30
Proof:
The Basic Differentiation Rules
𝑑
𝑑𝑥
𝑐𝑓 𝑥 = 𝑐𝑓′
𝑥
Substitution:
𝑓′ 𝑥 = lim
ℎ→0
𝑓 𝑥 + ℎ − 𝑓 𝑥
ℎ

31. 31
Proof:
Therefore,
𝑑
𝑑𝑥
𝑐𝑓 𝑥 = 𝑐𝑓′ 𝑥 .
The Basic Differentiation Rules

32. 32
Example:
Use the Constant Multiple Rule to find the derivative of
𝑓 𝑥 = 4𝑥6.
The Basic Differentiation Rules

33. 33
Example:
Use the Constant Multiple Rule to find the derivative of
𝑓 𝑥 = 4𝑥6.
The derivative of 𝑓 𝑥 = 4𝑥6 is 𝑓′ 𝑥 = 24𝑥5.
The Basic Differentiation Rules

34. 34
Sum or Difference Rule
Let 𝑓 and 𝑔 be differentiable functions, then the sum (or
difference) of 𝑓 and 𝑔 is also differentiable.
The derivative of the sum of 𝑓 and 𝑔 is given by
𝒅
𝒅𝒙
𝒇 𝒙 + 𝒈 𝒙 = 𝒇′ 𝒙 + 𝒈′ 𝒙 .
The Basic Differentiation Rules

35. 35
Sum or Difference Rule
The derivative of the sum of 𝑓 and 𝑔 is given by
𝒅
𝒅𝒙
𝒇 𝒙 − 𝒈 𝒙 = 𝒇′
𝒙 − 𝒈′
𝒙 .
The Basic Differentiation Rules

36. 36
Proof:
The Basic Differentiation Rules
𝑑
𝑑𝑥
𝑓 𝑥 + 𝑔 𝑥 = lim
ℎ→0
𝑓 𝑥 + ℎ + 𝑔 𝑥 + ℎ − 𝑓 𝑥 + 𝑔 𝑥
ℎ
Limit Definition of Derivative

37. 37
Proof:
The Basic Differentiation Rules
𝑑
𝑑𝑥
𝑓 𝑥 + 𝑔 𝑥 = lim
ℎ→0
𝑓 𝑥 + ℎ + 𝑔 𝑥 + ℎ − 𝑓 𝑥 − 𝑔 𝑥
ℎ
Distributivity

38. 38
Proof:
The Basic Differentiation Rules
𝑑
𝑑𝑥
𝑓 𝑥 + 𝑔 𝑥 = lim
ℎ→0
𝑓 𝑥 + ℎ − 𝑓 𝑥 + 𝑔 𝑥 + ℎ − 𝑔 𝑥
ℎ
Associativity

39. 39
Proof:
The Basic Differentiation Rules
𝑑
𝑑𝑥
𝑓 𝑥 + 𝑔 𝑥 = lim
ℎ→0
𝑓 𝑥 + ℎ − 𝑓 𝑥
ℎ
+ lim
ℎ→0
𝑔 𝑥 + ℎ − 𝑔 𝑥
ℎ
Sum or Difference Law for Limits

40. 40
Proof:
The Basic Differentiation Rules
𝑑
𝑑𝑥
𝑓 𝑥 + 𝑔 𝑥 = 𝑓′
𝑥 + 𝑔′(𝑥)
Limit Definition of Derivative

41. 41
Example:
Use the Sum or Difference Rule to find the derivative of
𝑓 𝑥 = 𝑥3 + 𝑥2.
The Basic Differentiation Rules

42. 42
Example:
Use the Sum or Difference Rule to find the derivative of
𝑓 𝑥 = 𝑥3 + 𝑥2.
The derivative of the function 𝑓 𝑥 = 𝑥3 + 𝑥2 is
𝑓′ 𝑥 = 3𝑥2 + 2𝑥.
The Basic Differentiation Rules

43. 43
What differentiation rules can
be used to get the derivative of
𝒇 𝒙 = 𝒙𝟑
+ 𝟑𝒙𝟐
+ 𝟓?

44. Let’s Practice!
44
Find the derivative of 𝒚 = 𝟒𝒙𝟕

45. Let’s Practice!
45
Find the derivative of 𝒚 = 𝟒𝒙𝟕
𝒚′
= 𝟐𝟖𝒙𝟔

46. Try It!
46
46
Find the derivative of 𝒚 = −𝟓𝒙𝟗
.

47. Tips
47
In differentiating functions of the
form 𝒇 𝒙 = 𝒄 ∙ 𝒙𝒌
, we can find the
derivative of the function by
multiplying the exponent to the
coefficient of the function and
subtracting 1 from the exponent.

48. Tips
48
To differentiate the function
𝒇 𝒙 = 𝟑𝒙𝟔, we can directly multiply
the exponent 6 to the coefficient 3
and subtract 1 from the exponent.
Thus, the derivative of the function
will be 𝒇′
𝒙 = 𝟏𝟖𝒙𝟓
.

49. Let’s Practice!
49
What is the derivative of the function
𝒇 𝒙 = 𝒙𝟑 + 𝟑𝒙𝟐 − 𝟔𝒙 + 𝟖

50. Let’s Practice!
50
What is the derivative of the function
𝒇 𝒙 = 𝒙𝟑 + 𝟑𝒙𝟐 − 𝟔𝒙 + 𝟖
𝒇′
𝒙 = 𝟑𝒙𝟐
+ 𝟔𝒙 − 𝟔

51. Try It!
51
51
What is the derivative of the function
𝒇 𝒙 = 𝟐𝒙𝟑 − 𝟗𝒙𝟐 + 𝟒𝒙 − 𝟔?

52. Let’s Practice!
52
Find the equation of the line tangent to the curve
𝒚 = 𝒙𝟑 − 𝒙𝟐 at 𝒙 = 𝟏.

53. Let’s Practice!
53
Find the equation of the line tangent to the curve
𝒚 = 𝒙𝟑 − 𝒙𝟐 at 𝒙 = 𝟏.
𝒚 = 𝒙 − 𝟏 or 𝒙 − 𝒚 = 𝟏

54. Try It!
54
54
Find the equation of the line tangent to
the curve 𝒚 = 𝒙𝟒 − 𝒙𝟐 at 𝒙 = −𝟏.

55. Let’s Practice!
55
Find the instantaneous rate of change in
𝒇 𝒙 = 𝟑𝒙𝟑 + 𝒙𝟐 − 𝟓 with respect to 𝒙 at the instant
when 𝒙 = 𝟏.

56. Let’s Practice!
56
Find the instantaneous rate of change in
𝒇 𝒙 = 𝟑𝒙𝟑 + 𝒙𝟐 − 𝟓 with respect to 𝒙 at the instant
when 𝒙 = 𝟏.
𝟏𝟏

57. Try It!
57
57
Find the instantaneous rate of change in
𝒇 𝒙 = 𝟐𝒙𝟑
− 𝟒𝒙𝟐
− 𝟗 with respect to 𝒙 at
the instant when 𝒙 = −𝟐.

58. Remember
58
The first derivative of the function 𝒇 𝒙
tells us about the instantaneous rate of
change when 𝒙 = 𝒄, where 𝒄 is a constant.
● If 𝒇′ 𝒙 > 𝟎, the function is increasing
at the instant when 𝒙 = 𝒄.
● If 𝒇′ 𝒙 < 𝟎, the function is decreasing
at the instant when 𝒙 = 𝒄.

59. Check Your Understanding
59
Find the derivative of each function.
1. 𝑓 𝑥 = 26
2. 𝑦 = 4 − 2𝑥
3. 𝑔 𝑥 = −9𝑥18
4. 𝑓 𝑥 =
1
𝑥3
5. 𝑦 = 𝑥2
+ 𝑥 + 1 𝑥 − 1

60. Check Your Understanding
60
Solve the following problems.
1. What is the instantaneous rate of change for the
function 𝑦 = 3𝑥2 − 𝑥 − 2 when 𝑥 = 2?
2. On which points on the curve 𝑦 = 𝑥3
− 3𝑥2
+ 4 are the
tangent lines horizontal?

61. Let’s Sum It Up!
61
The following are the Basic Differentiation Rules,
which were all derived from the limit definition of
derivatives.
● Constant Rule
𝑑
𝑑𝑥
𝑐 = 0

62. Let’s Sum It Up!
62
● Power Rule
𝑑
𝑑𝑥
𝑥𝑘
= 𝑘𝑥𝑘−1

63. Let’s Sum It Up!
63
● Constant Multiple Rule
𝑑
𝑑𝑥
𝑐𝑓 𝑥 = 𝑐 ∙ 𝑓′
𝑥

64. Let’s Sum It Up!
64
● Sum or Difference Rule
𝑑
𝑑𝑥
𝑓 𝑥 + 𝑔 𝑥 = 𝑓′
𝑥 + 𝑔′
𝑥
𝑑
𝑑𝑥
𝑓 𝑥 − 𝑔 𝑥 = 𝑓′
𝑥 − 𝑔′
𝑥

65. Key Formulas
65
Concept Formula Description
Constant Rule 𝑑
𝑑𝑥
𝑐 = 0
Use this formula to
solve for the derivative
of a constant function.
Power Rule 𝑑
𝑑𝑥
𝑥𝑘
= 𝑘𝑥𝑘−1
Use this formula to
solve for the derivative
of a variable raised to a
real number exponent.

66. Key Formulas
66
Concept Formula Description
Constant
Multiple Rule
𝑑
𝑑𝑥
𝑐𝑓 𝑥 = 𝑐𝑓′
𝑥
Use this formula to
solve for the derivative
of a constant times a
function.
Sum or
Difference Rule
𝑑
𝑑𝑥
𝑓 𝑥 + 𝑔 𝑥 = 𝑓′ 𝑥 + 𝑔′
𝑥
𝑑
𝑑𝑥
𝑓 𝑥 − 𝑔 𝑥 = 𝑓′ 𝑥 − 𝑔′
𝑥
Use this formula to
solve for the derivative
of sum or difference of
functions.

67. Challenge Yourself
67
67
In the function 𝒇 𝒙 = 𝒂𝒙𝟐
− 𝒙 − 𝟏 ,
what value of 𝒂 will make 𝒇′ 𝟐 = 𝟕?

68. Bibliography
68
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.