BARU CHAPTER 2 (2.2 - 2.3 trigo, log, exp , second order) pdf

CHAPTER 2:
DIFFERENTIATION

Introduction

Expression: when differentiate, we write as:



Equation: y = when differentiate, we write as:



Function: = when differentiate, we write as: ′

a) Mesti ada KUASA = 5 3 3
b) KUASA mesti di ATAS 2
= 5
= 4 = 2 −4

c) 0 = 1 = 2 − − 2
2 5 2 5
d) Expand = + 2 1 −
d) Separate 2 5 − 2 + 5 = 2 − 2 + 2
= 2 3 − 1 + 5 −2
= 2

TRIGONOMETRIC
FUNCTION

CHAPTER 2: DIFFERENTIATION

Trigonometric Function

Reciprocal Identities Trigonometric Identities

2 + 2 = 1
= 1 + 2 = 2
2 + 1 = 2
1
=

1
=

1
= =

Trigonometric Function

Basic Derivatives


= = −


= − =

= 2 = − 2


Trigonometric Function

How To Derive Trigonometric Function

1. Differentiate trigo
= ∙ 1 2. Differentiate

1. Differentiate trigo
+ = + ∙ 2. Differentiate dlm kurungan

+ = −1 + ∙ + ∙


1. Differentiate KUASA

2. Differentiate trigo

3. Differentiate dlm kurungan

Trigonometric Function 1. Differentiate trigo
2. Differentiate

Example 7: Find the derivative
= ∙ 1

a) = 3 c) = 2 d) = 3 2 kuasa bawa
8 4 naik ATAS
3
= − 3 ∙ 3 = 3 2 −4
2 2 8
= −3 3 = 3 ∙ 3
3 − 2 −4 ∙ −8 −5
22 = 8
= 3 3

b) = 1 4 2
= 3 5
2

= 1 24 ∙ 4
2

= 2 2 1 − 4

Trigonometric Function 1. Differentiate trigo
2. Differentiate dlm kurungan

Example 7: Find the derivative = + ∙
+

a) = 3 + 2 b) = 5 + 3

= − 5 + 3 5 + 3 2
= − 3 + 2 3

= −3 3 + 2 = − 5 + 3 2 5 + 3

c) = 3 1 − 4 3 composite d) = 7 − 2
function 4
8
tukar btk
= 7 − 2 −4
= 3 1 − 4 3 3 1 − 4 2 −4
8
= 2 7 − 2 −4 −8 −5

9 1 − 4 2 1 − 4 3 = − 8 2 2
= −2 5 7 − 4

Trigonometric Function 1. Differentiate KUASA
2. Differentiate trigo

3. Differentiate dlm kurungan

Example 7: + = −1 + ∙ + ∙


a) = 34 b) = 2 2 5 + 3

= 3 ∙ 24 ∙ −sin 4 4 = 2∙ 2 5 + 3 − 5 + 3 5 + 3 2


= −3 24 4 = −4 5 + 3 2 5 + 3 5 + 3

c) = 3 5 1 − 4 d) = 4 7 − 1 kuasa bawa
4 naik ATAS
5

= 3 ∙ 5 4 1 − 4 ∙ 1 − 4 −4 = 4 7 − −4
5
= 4 3 7 − −4 2 7 − −4 −4 −5

= −12 4 1 − 4 ∙ 1 − 4 = − 16 3 7 − −4 2 7 1
5 − 4

Trigonometric Function

Example 7: Find the derivative

a) = 4 5 − 4 2 Product Rule:

= 4 = 5 − 4 2 = +

= 4 3 = 5 − 4 2 −8


= −8 5 − 4 2

= 5 − 4 2 ∙ 4 3 + 4 ∙ −8 5 − 4 2

= 4 3 5 − 4 2 − 2 2 5 − 4 2

Trigonometric Function

Example 7: Find the derivative

b) = 2+5 Quotient Rule:
3+2 2

= 2 + 5 = 3 + 2 2 = −

2

= − 2 +5 ∙ 2 = 3 2 + 4


= −2 2 + 5

3 + 2 2 ∙ −2 2 + 5 − 2 + 5 ∙ 3 2 + 4
=
( 3 + 2 2)2

−2 3 + 2 2 2 + 5 − 3 2 + 4 2 + 5
= ( 3 + 2 2)2

Trigonometric Function

Example 7: Find the derivative

c) = 3 3 Chain Rule:

= 3 = 3 = ×

= 2 3(3 2) = 2 2


= 3 2 2 3

= 2 2 ∙ 3 2 2 3


= 6 2 2 3 2

= 6 2 2 3 2 3 3

LOGARITHMIC
FUNCTION

CHAPTER 2: DIFFERENTIATION

Logarithmic Function

How To Derive Logarithmic Function

If given :

a) ln b) ln c) ln

1 ′ 1
∴ ln = ∴ ln = ∴ ln = ∙

Example: One over Differentiate
value value

a) ln 2 b)Differentiate ln 2 Over value c) ln 2
value
1 1 2 1 1 2 1
= 2 ∙ 2 = = 2 = = 2 ∙ =

a) + =

b) − =


c) =

d) = 1

e) =


Logarithmic Function

Example 8: Find the derivative

Hukum log:

=

a) = ln 4 differentiate b) = ln 3 5 c) = ln 3 − 2
constant
= 5 ln 3 1
= 0 = 3 − 2 3
1
= 5 ∙ 3 ∙ 3 3
= 3 − 2
5
=

Logarithmic Function

Example 8: Find the derivative

Hukum log:

=

d) = ln 1 − 3 5 e) = ln 5 + 2 tukfa)r = 3 ln 2 − 3 + 5 2
1
= 5 ln 1 − 3 bentuk 1 −3 + 10
= ln 5 + 2 2 = 3 ∙ 2 − 3 + 5 2
1 1
= 5 ∙ 1 − 3 ∙ −3 = 2 ln 5 + 2
3 −3 + 10
−15 = 2 − 3 + 5 2
= 1 − 3 1 1
= 2 ∙ 5 + 2 5

5
= 2 5 + 2

Logarithmic Function

Example 8: Find the derivative

Hukum log: Hukum log:

= ln 5 − 3 2 3 2 + 4 2 = 2 + 2 7 2 = 2 − 2
g) h) = ln 1−3 2

bw kuasa = ln 5 − 3 2 + ln 3 2 + 4 = ln 7 − ln 1 − 3 2

turun = 2 ln 5 − 3 + ln 3 2 + 4 1 1
= 7 7 − 1 − 3 2 −6
1 1
′ =2∙ 5 − 3 ∙5+ 3 2 + 4 ∙ 6 1 6
= 7 + 1 − 3 2
10 6
= 5 − 3 + 3 2 + 4

Logarithmic Function Quotient Rule:

= −


j) = ln 5 +2 2
4
Example 8: Find the derivative = 4

i) = 4 ln 5 + 2 Product Rule: = ln 5 + 2 = 4 3
1
= + = 5 + 2 5

= 4 = ln 5 + 2 5
= 5 + 2
1
= 4 3 = 5 + 2 5 4 ∙ 5 − ln 5 + 2 ∙ 4 3
= 5 + 2 4 3 2
5 ′
= 5 + 2
5
′ = ln 5 + 2 ∙ 4 3 + 4 ∙ 5 3 5 + 2 − 4 ln 5 + 2
=
5 + 2 4 6

= 3 4 ln 5 + 2 + 5 = 5 − 4 ln 5 + 2
5 + 2 5 + 2 4 3

Logarithmic Function

Example 8: Find the derivative

i) = ln sin 3 2 j) = 2

′ = 1 ∙ cos sin 3 2 ∙ 6 ′ = 2 2 1
sin 3 2 ∙ 2 ∙ 2

6 cos sin 3 2 2 2
= sin 3 2 =

Logarithmic Function

= 2 − 3 2 − 3 Hukum log: = 5 − 3 2 Hukum log:
=
2 = 2 + 2

= 2 − 3 2 − 3 Product Rule: = 5 − 3 2 Chain Rule:

= +
= ×

7 Hukum log:
= 1 − 3 2

2 = 2 − 2

7 Quotient Rule:
= 2 − 3 2
= −


2

EXPONENTIAL
FUNCTION

CHAPTER 2: DIFFERENTIATION

Exponential Function

How To Derive Exponential Function c) +

If given : ∴ + = + ∙ +

a) b)

∴ = ∴ = ∙ ′ c) 2 +1 Salin semula
exponent

Salin semula Salin semula = 2 +1 ∙ 2
exponent
Example: exponent

a) 2 Differentiate b) 2 Differentiate = 2 2 +1 Differentiate
power power power

= 2 ∙ 2 = 2 ∙ 2

= 2 2 = 2 2

a) × = + f) =



b) ÷ = − g) 0 = 1

c) =

d) 2 = 2 2

e) − = 1


Exponential Function e) = 5
3 2
Example 9: Find the derivative

c) = 3 5 = 5 −2
3
a) = 5
differentiate = 5 −2 ∙ −2
3
= 0 constant = 3 ∙ 5 ∙ 5

= − 10 −2
3
= 15 5

b) = 5 d) = 5 +2 f) = 2 3 3

= 5 ∙ 5 5 +2 = 8 9

= ∙ 5 = 8 9 ∙ 9

= 5 5 = 5 5 +2
= 72 9

Exponential Function

Example 9: Find the derivative

Hukum index Hukum index
Hukum index
× = + 5 −2 ÷ = − =
3 2
g) = 2 +3 5 3 h) i) 4 16 8
= =

= 5 2 +3+3 1 = 4 16 ∙ 8
3
= 5 −2−2 4

= 5 5 +3 1 = 4 2
3
= 5 5 +3 5 = 3 −2 = 4 2 ∙ 2

1
= 25 5 +3 = 3 3 −2 ∙ 3 = 8 2

= 3 −2

Exponential Function

Example 9: Find the derivative

j) = 2 3 − 6 expand k) = 5 + l) = 6 − 2 tukar
2 2 bentuk
separate
= 5 − 6 2 5 = 6 − 2 −2
= 2 + 2
5 − 6 2 = 6 ∙ 6 − 2 −2 ∙ −2
= 5 2 = 3 + −

= 5 5 − 12 2 = 3 3 + − −1 = 6 6 + 4
2

= 3 3 − 1


Exponential Function

Example 9: Find the derivative

m) = 4

Method 1: Chain Rule Method 2: Exponential Function

= 4 = = 4 ∙ 4 3

4 3
= = = 4 3 4

= ∙ 4 3 NOTE PENTING:
u ialah KUASA bg
exponent tsbt !!
= 4 3 4

Exponential Function

Example 9: Find the derivative (Chain Rule/ Exponential Function)

n) = 2

Method 1: Chain Rule Method 2: Exponential Function

= 2 = 1 2 2
= 2 ∙
2 2 1
= = =2

1 2 2
= ∙

2 2
=

2 2 =2
= 2

Exponential Function

Example 9: Find the derivative (Chain Rule/ Exponential Function)

o) = 4 − 2
1
= 4 − 2 2

Method 1: Chain Rule Method 2: Exponential Function

= 4 − 2 1 1 4 − 2 −21 ∙ −2 2
= 2
= 2

= −2 2 = 1 −21 2
2 =

1 −12 4 − 2
2
= ∙ −2 2

2 2
=1 =

2 4 − 2

Exponential Function Quotient Rule:

ln 5 +2 = −
4
q)
Example 9: Find the derivative = 2

Product Rule: = ln 5 + 2 = 4

p) = 2 2 + 1 3 = + 1 = 4 3
= 5 + 2 5

= 2 = 2 + 1 3 5
= 5 + 2

= 2 2 = 3 2 + 1 2 2 4 ∙ 5 ∙ 4 3
= 5 + 2
= 6 2 + 1 2 ′ − ln 5 + 2
4 3 2

′ = 2 + 1 3 ∙ 2 2 + 2 ∙ 6 2 + 1 2 3 5 − 4 ln 5 + 2
= 5 + 2
= 2 2 2 + 1 2 2 + 1 + 3
= 2 2 2 + 1 2 2 + 1 + 3 5 4 6
5 + 2
= − 4 ln 5 + 2
4 3

Logarithmic Function

One over

value

ln 2 Differentiate

How to differentiate these function? 1 value
= 2 ∙ 2
Trigonometric Function
1
=
+ = + ∙
Exponential Function
1. Differentiate trigo
2. Differentiate dlm kurungan

Salin semula
exponent

+ = −1 + ∙ + ∙ 2 Differentiate
power

1. Differentiate KUASA = 2 ∙ 2
2. Differentiate trigo
= 2 2
3. Differentiate dlm kurungan

SECOND ORDER
DIFFERENTIATION

CHAPTER 2: DIFFERENTIATION

Second Order Differentiation

What is Second Order Differentiation?

In calculus, the second derivative, or the second order differentiation,
of a function is the derivative of the derivative of .

2
Written as:- ′′ or 2

differentiate differentiate differentiate differentiate
first time second time first time second time

′ ′′ 2

2

a) Mesti ada KUASA = 5 3 3
b) KUASA mesti di ATAS 2
= 5
= 4 = 2 −4

c) 0 = 1 = 2 − − 2
2 5 2 5
d) Expand = + 2 1 −
d) Separate 2 5 − 2 + 5 = 2 − 2 + 2
= 2 3 − 1 + 5 −2
= 2

Second Order Differentiation

Example 10: Find the second derivative for each of the following:

a) = 3 3 − 2 2 + 7 b) = 2 5 − 4 3 + 2 − 7 tukar

First 7
derivative =
3 3 2 − 2 2 First 2 5 3 2 −1 bentuk
derivative 7
= − 4 + − 7

Second = 9 2 − 4 5 2 4 − 3 −14 + 2 −1 −2
derivative 2 = 4 7
2 = 2 9 − 4
Second = 10 4 − 3 −14 − 2 −2
derivative 4 7
= 18 − 4
2 1 3 −54 2
2 = 4 10 4 − −4 4 − −2 7 −3

= 40 4 + 3 5 + 4
16 4 7 3

Second Order Differentiation

Example 10: Find the second derivative for each of the following:

c) = 2 3 + 2 tukar First d) = 3 2 + 3
3 bentuk derivative 1
First
derivative = 2 3 + 2 −3 = 3 ∙ 2 + 3 ∙ 2

+ −3 2 −4 Second 6
= 2 3 3 derivative = 2 + 3
Second
derivative = 6 3 − 6 −4 = 6 2 + 3 −1

2 3 − −4 6 −5 2 2 + 3 −2 2
2 = 6 − 3 2 =
−1 6
= −18 3 + 24 −5
−12
24 = 2 + 3 2
= −18 3 + 5

Second Order Differentiation

Example 10: Find the second derivative for each of the following:

e) = 4 3 +1 + 2 − 10 tukar First f) = 3 + 5 4
5
First bentuk derivative
= 4 ∙ 3 + 5 3 ∙
derivative = 4 3 +1 + 2 −5 − 10 3

= 4 3 +1 3 + 2 −5 −5 Second = 12 3 + 5 3
derivative
Second 2
derivative = 12 3 +1 − 10 −5 2 = 3 12 3 + 5 2 3

2 = 12 3 +1 3 − 10 −5 −5 = 108 3 + 5 2
2

= 36 3 +1 + 50 −5

= 36 3 +1 + 50
5

Second Order Differentiation

Example 10: Find the second derivative for each of the following:

g) = 2 3 Product Rule: = 6 = 3

= 6
= + = 3 3

= 2 = 3 = 3 3

= 2 ′′ = 2 − 3 3 − 3 6 + 6 3 3
= − 3 3 = −6 3 − 6 3 − 18 3

= −3 3

′ = 3 2 + 2 −3 3

= 2 3 − 6 3 = −12 3 − 18 3

= −6 2 3 − 3 3

Second Order Differentiation

Example 10: Find the second derivative for each of the following:

Quotient Rule:

h) = 2+3 = − = 4 2 − 2 − 12 = 4 − 1 2

4 −1 = 8 − 2 = 8 4 − 1
2

= 2 + 3 = 4 − 1 2 4 − 1 2 8 − 2 − 4 2 − 2 − 12 ∙ 8 4 − 1
2 =
= 2 = 4 4 − 1 2 2

4 − 1 2 − 2 + 3 ∙ 4 4 − 1 4 − 1 8 − 2 − 8 4 2 − 2 − 12
= = 4 − 1 4
4 − 1 2

8 2 − 2 − 4 2 − 12 36 2 − 4 − 8 + 2 − 36 2 + 16 + 96
= 4 − 1 2 = 4 − 1 3

4 2 − 2 − 12 4 + 98
= 4 − 1 2 = 4 − 1 3

Second Order Differentiation

Example 10: Find the second derivative for each of the following:

i) = Quotient Rule:

+2 −
2
=

= = + 2

= 1 = 1

′ = + 2 1 − 1 ′ = 2 + 2 −2
′′ = −2 ∙ 2 + 2 −3 1
+ 2 2
= −4 + 2 −3
+ 2 −
= + 2 2 −4

2 = + 2 3
= + 2 2

Second Order Differentiation

Example 10: Find the value of 2 when = −2 for each of the following
2

a) = 3 − 6 2 + 9 + 3 b) = + 1 3 2 − 1

= 3 2 − 12 + 9 = 3 3 − + 3 2 − 1

9 2
2 = − 1 + 6
2 = 6 − 12
2
when = −2 2 = 18 + 6

2 when = −2
∴ 2 = 6 −2 − 12
2
= −24 ∴ 2 = 18 −2 + 6

= −30