BASIC CALCULUS FOR STEM 11

1. RULES OF DIFFERENTIATION
PART 2

2. GOOD
AFTERNOON
MS. JESSEL 0RENCIO
2

3. GOALS FOR THIS WEEK
I AM ABLE TO:
● derive the differentiation rules;
● apply the differentiation rules in computing the derivatives
1. algebraic functions,
2. exponential functions,
3. logarithmic functions, and
4. trigonometric functions;
● display perseverance to solve for the derivatives of functions.
3

4. CHAIN RULE
𝑑
𝑑𝑥
𝑢𝑛
= 𝑛𝑢𝑛−1
𝑑𝑢
𝑑𝑥
EXAMPLE 1. Find the derivative of 𝑦 = 𝑥2 + 5
𝑦 = 𝑥2 + 5 1/2

5. CHAIN RULE
EXAMPLE 1. Find the derivative of 𝑦 = 𝑥2 + 5
𝑦 = 𝑥2
+ 5 1/2
Let 𝑢 = 𝑥2
+ 5
𝑛 = 1/2
𝑑𝑦
𝑑𝑥
=
1
2
𝑥2 + 5 −
1
2
𝑑
𝑑𝑥
(𝑥2 + 5)
𝑑𝑦
𝑑𝑥
=
1
2
𝑥2 + 5 −1/2 2𝑥
𝑑𝑦
𝑑𝑥
=
𝑥
𝑥2+5 1/2
𝑑𝑦
𝑑𝑥
=
𝑥
𝑥2+5

6. CHAIN RULE
EXAMPLE 2. Find the derivative of 𝑦 = 𝑥 + 5 2
𝑦 = 𝑥 + 5 2
Let 𝑢 = 𝑥 + 5
𝑛 = 2
𝑑𝑦
𝑑𝑥
= 2 𝑥 + 5 2−1
𝑑
𝑑𝑥
𝑥 + 5
𝑑𝑦
𝑑𝑥
= 2(𝑥 + 5)(1)
𝑑𝑦
𝑑𝑥
= 2𝑥 + 10

7. Differentiation of
Exponential and
Logarithmic Functions
LESSON 7.2
7

8. The next set of functions that we like to focus on are
exponential and logarithmic functions. The most commonly
used exponential form in a calculus course is the natural
exponential function 𝒆𝒙
.
8
Summary of Derivative of
Exponential Functions

9. Solution:
𝑑
𝑑𝑥
= 𝑎𝑥
= 𝑎𝑥
ln 𝑎
𝑑
𝑑𝑥
= (𝑥𝟑
+ 𝟑𝑥
) = 𝟑𝑥𝟐
+ 𝟑𝑥
(ln 𝟑)
EXAMPLE 1.
Find the derivative of 𝑦 = 𝑥3 + 3𝑥 using
the differentiation rule.
Formula
Solved the derivative of 𝑥𝟑
and 𝟑𝑥
.

10. Solution:
𝑑
𝑑𝑥
𝑒𝑢 = 𝑒𝑢 𝑑𝑢
𝑑𝑥
𝑑𝑦
𝑑𝑥
= 𝑒𝑒𝑥
= 𝑒𝑒𝑥 𝑑
𝑑𝑥
𝑒𝑥
= 𝑒𝑒𝑥
∙ 𝑒 ∙
𝑑
𝑑𝑥
(𝑥)
= 𝑒𝑒𝑥 ∙ 𝑒 ∙ (1)
= 𝑒𝑒𝑥+1
EXAMPLE 2.
Find the derivative of 𝑦 = 𝑒𝑒𝑥.
10
Let 𝒖 = 𝒆𝒙 𝒅𝒖 = 𝒆
Formula
Differentiation is linear. Differentiated them
separately and pulled out constant factors
Derivative of x is 1

11. EXAMPLE 3.
Differentiate 𝑦 = 𝑥 ∙ ln 𝑥 − 𝑥
11
Solution:
𝑑𝑦
𝑑𝑥
= 𝑥 ∙ ln 𝑥 − 𝑥
=
𝑑
𝑑𝑥
𝑥 ∙ ln 𝑥 −
𝑑
𝑑𝑥
(𝑥)
=
𝑑
𝑑𝑥
𝑥 ∙ ln 𝑥 + 𝑥 ∙
𝑑
𝑑𝑥
ln 𝑥 − 1
= 1 ∙ ln 𝑥 + 𝑥 ∙
1
𝑥
− 1
= ln 𝑥
Let 𝒖 = 𝒙 𝒅𝒖 = 𝟏
𝒗 = 𝐥𝐧 𝒙 𝒅𝒗 =
𝟏
𝒙
(to be discussed in logarithm)
Differentiation is linear. Differentiated
them separately and pulled out constant
factors
Apply the product rule
Simplified (derivative of 𝒍𝒏𝒙 is
𝟏
𝒙
)
Final answer

12. EXAMPLE 4. Find the derivative of 𝑦 = 𝑒4𝑥+7.
12
Solution:
𝑑
𝑑𝑥
𝑒𝑢
= 𝑒𝑢 𝑑𝑢
𝑑𝑥
= 𝑒4𝑥+7 𝑑
𝑑𝑥
(4𝑥 + 7)
= 𝑒4𝑥+7 4 ∙
𝑑
𝑑𝑥
𝑥 +
𝑑
𝑑𝑥
(7)
= 𝑒4𝑥+7
4 ∙ 1 + 0
= 𝟒𝒆𝟒𝒙+𝟕
Let 𝒖 = 4𝑥 + 7 𝒅𝒖 = 𝟒
Formula
Differentiation is linear. Differentiated them
separately and pulled out constant factors
Differentiated each term
Final answer

13. Differentiating a
Logarithmic Function

14. Expressions written in exponential form can be converted
to logarithmic function and vice versa.
Exponential Form to Logarithmic Form
53
= 125 ⟹ log5 125 = 3
490.5
= 7 ⟹ log49 7 = 0.5
Logarithmic Form to Exponential Form
log2 8 = 3 ⟹ 23
= 8
log3 81 = 4 ⟹ 34
= 81
Hence 𝒚 = 𝐥𝐨𝐠𝒃 𝒙 can be written as 𝑏𝑦 = 𝑥 and
𝒚 = 𝐥𝐨𝐠𝒆 𝒙 can be written as 𝑒𝑦 = 𝑥.
• natural logarithms are to the base 𝒆
• 𝐥𝐧 𝒙 is used for natural logarithms

15. The derivative of the Natural Logarithm Function
If 𝑦 = ln 𝑥, then
𝑑
𝑑𝑥
ln 𝑥 =
1
𝑥
.
15
If 𝑢 is a differentiable function of 𝑥, then according to the Chain Rule:

16. Derivative of Logarithmic Functions other than the
natural logarithms.
𝑑
𝑑𝑥
(log𝑏 𝑥) =
1
𝑥 ln 𝑏
.
16
If 𝑢 is a differentiable function of 𝑥, then

17. EXAMPLE 1. Differentiate 𝑦 = ln 5𝑥
17
Solution: Use
𝑑
𝑑𝑥
ln 𝑢 =
1
𝑢
∙
𝑑𝑢
𝑑𝑥
𝑦 = ln(5𝑥)
𝑑𝑦
𝑑𝑥
=
1
5𝑥
∙
𝑑
𝑑𝑥
(5𝑥)
𝑑𝑦
𝑑𝑥
=
1
5𝑥
5
𝑑𝑦
𝑑𝑥
=
5
5𝑥
=
1
𝑥
Let 𝒖 = 𝟓𝒙 𝒅𝒖 = 𝟓

18. EXAMPLE 1. Differentiate 𝑦 = ln 5𝑥
18
𝑦 = ln 5𝑥
𝑑𝑦
𝑑𝑥
=
1
𝑥

19. EXAMPLE 2.Find the derivative of 𝑦 = ln(𝑥3 + 4).
19
Solution: Use
𝑑
𝑑𝑥
ln 𝑢 =
1
𝑢
∙
𝑑𝑢
𝑑𝑥
𝑦 = ln(𝑥3 + 4)
𝑑𝑦
𝑑𝑥
=
1
𝑥3+4
∙
𝑑
𝑑𝑥
(𝑥3 + 4)
𝑑𝑦
𝑑𝑥
=
1
𝑥3+4
3𝑥2
𝑑𝑦
𝑑𝑥
=
3𝑥2
𝑥3+4
Let 𝒖 = 𝑥3 + 4 𝒅𝒖 = 𝟑𝒙𝟐

20. 20
𝑑𝑦
𝑑𝑥
=
3𝑥2
𝑥3 + 4
𝑦 = ln 5𝑥
EXAMPLE 2.Find the derivative of 𝑦 = ln(𝑥3 + 4).

21. Differentiation of
Trigonometric
Functions
LESSON 7.3
21

22. EXAMPLE.
Differentiate 𝑦 = sin 4𝑥
Chain rule has been applied here.
Differentiation is linear.
Differentiated them separately and pulled
out constant factor
Final answer

23. EXAMPLE.
Differentiate 𝑦 = cos(2𝑥)
Chain rule has been applied here.
Differentiation is linear.
Differentiated them separately and pulled
out constant factor
Derivative of 𝒙 is 1.
Final answer

24. EXAMPLE.
Differentiate 𝑦 = tan(2𝑥)
Chain rule has been applied here.
Differentiation is linear.
Differentiated them separately and pulled
out constant factor
Derivative of 𝒙 is 1.
Final answer

25. EXAMPLE.
Differentiate 𝑦 = 4𝑥2 +cot 𝑥
Differentiation is linear.
Differentiated them separately
and pulled out constant factor.
Applied the differentiation rule
for 𝑐𝑜𝑡 𝑥 and applied the power
rule for 𝑥𝟐.
Final answer

26. EXAMPLE.
Differentiate 𝑦 = sec(2𝑥)
Differentiation is linear.
Differentiated them separately
and pulled out constant factor.
Differentiated them separately
and pulled out constant factor.
Derivative of x is 1.
Final answer

27. EXAMPLE.
Differentiate 𝑦 = csc 5𝑥
Differentiation is linear.
Differentiated them separately
and pulled out constant factor.
Differentiated them separately
and pulled out constant factor.
Derivative of x is 1.
Final answer

28. SUMMARY

29. Derivative of Exponential and
Logarithmic Functions Exponential Functions
Logarithmic Functions
𝒅
𝒅𝒙
𝒍𝒏 𝒙 =
𝟏
𝒙

30. Rules of Derivatives of
Trigonometric Functions
RULE 1 RULE 2
RULE 3

31. Rules of Derivatives of
Trigonometric Functions
RULE 4
RULE 5
RULE 6

32. 32
NICE job for
LISTENING
WELL!
Ask me question/s…
Message me @:
❑ messenger: Jessel-Ann Orencio Lpt
❑ email: jessel-ann.orencio@deped.gov.ph

33. REFERENCES
❑ Basic Calculus – Grade 11 Alternative Delivery Mode
Quarter 3 – Module 7: Rules of Differentiation, First
Edition, 2020 Department of Education
❑ Department of Education (2016). Teaching Guide for
Senior High School Basic Calculus
❑ Differential Calculus Technological University of the
Philippines

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