1. DIFFERENTIATION
AND
APPLICATIONS

2. A brief introduction to differentiation Basic differentiation
CONTENTS
A brief introduction to differentiation
Basic differentiation
Differentiation using first principle
4. Product rule and Quotient rule
5. Chain rule
6. Implicit differentiation
7. Logarithmic differentiation
8. Higher derivatives
9. Differentiation of Parametric equations
10. Applications .

3. A brief introduction to differentiation Basic differentiation
CONTENTS
A brief introduction to differentiation
Basic differentiation
Differentiation using first principle
4. Product rule and Quotient rule
5. Chain rule
6. Implicit differentiation
7. Logarithmic differentiation
8. Higher derivatives
9. Differentiation of Parametric equations
10. Applications . 3

4. TECHNIQUES OF DIFFERENTIATION
Derivative is one of the fundamental idea in calculus.
Process of finding the derivatives of functions is called
differentiation.
By implementing the concept of limits, the derivative of
f (x) is given by
provided that the limit exists. .
This is known as differentiation of function f (x) by first
principle.

5. TECHNIQUES OF DIFFERENTIATION
Basic differentiation

6. Differentiation using first principle
Given y= f(x). Write the expression f(x + δx)
Obtain the expression difference between
f(x + δx) and f(x)
 f(x + δx) - f(x)
3. Simplify the expression
f(x + δx) - f(x) δx
Find the limit

7. EXAMPLE: pg 57
Given
STEP 1
STEP 2
STEP 3

8. EXAMPLE: pg 57
STEP 4
Let’s TRY!
Answer:

9. A brief introduction to differentiation Basic differentiation
Differentiation using first principle
4. Product rule and Quotient rule
5. Chain rule
6. Implicit differentiation
7. Logarithmic differentiation
8. Higher derivatives
9. Differentiation of Parametric equations
10. Applications . 9

10. PRODUCT & QUOTIENT RULE
Product rule
Quotient rule
Let
Let
Then
Then
Example: Product and quotient rule
(a) (a)
(b) (b)

11. A brief introduction to differentiation Basic differentiation
Differentiation using first principle
4. Product rule and Quotient rule
5. Chain rule
6. Implicit differentiation
7. Logarithmic differentiation
8. Higher derivatives
9. Differentiation of Parametric equations
10. Applications . 11

12. CHAIN RULE Chain rule allows the derivative to be solve without
the need to expand.
How do you know when to use the chain rule?
Look for parentheses!!
Bring the exponents to the front
Subtract exponent by 1
Multiply by inside derivative

13. EXAMPLE
STEP 1
STEP 2
STEP 3

14. LET’s TRY!
(a)
(b)
Answer:
(a)
(b)

15. A brief introduction to differentiation Basic differentiation
Differentiation using first principle
4. Product rule and Quotient rule
5. Chain rule
6. Implicit differentiation
7. Logarithmic differentiation
8. Higher derivatives
9. Differentiation of Parametric equations
10. Applications . 15

16. IMPLICIT DIFFERENTIATION
Compare the following two equations 1 2
Define y as a function of x
explicitly
Not Define y as a function
of x
The equation gives an
explicit formula y = f (x)
However, we can solve for
y and find at least two
functions that are defined
implicitly.
The differentiation of
implicit function is called
implicit differentiation

17. IMPLICIT DIFFERENTIATION
How to differentiate implicit functions??
TECHNIQUE 1
Let , example of implicit function. Find .
Step 1
Differentiate both side with respect to x
Step 2
Be careful to recognize that differentiating any
function of y will require the chain rule

18. IMPLICIT DIFFERENTIATION
Step 3
Gather on one side of the equation, then
solve for
Example: Implicit differentiation

19. LET’s TRY!
(Page )
(a)
(b)
(c)
Answer:
(c)
(a)
(b)

20. A brief introduction to differentiation Basic differentiation
Differentiation using first principle
4. Product rule and Quotient rule
5. Chain rule
6. Implicit differentiation
7. Logarithmic differentiation
8. Higher derivatives
9. Differentiation of Parametric equations
10. Applications . 20

21. LOGARITHMIC DIFFERENTIATION
Logarithmic differentiation is for a function:
1. That involve very tedious quotient function
Example:
2. Functions in form
Example:
Step 1
Taking natural logarithm (ln) of both sides
Step 2
Differentiate both side with respect to x and solve
For f ‘ (x)
Example: Logarithmic differentiation

22. LOGARITHMIC DIFFERENTIATION
Example: Logarithmic differentiation
Differentiate
(1)
(2)

23. LET’s TRY!
(Page 104)
(a)
(b)
Answer:
(a)
(b)

24. A brief introduction to differentiation Basic differentiation
Differentiation using first principle
4. Product rule and Quotient rule
5. Chain rule
6. Implicit differentiation
7. Logarithmic differentiation
8. Higher derivatives
9. Differentiation of Parametric equations
10. Applications . 24

25. HIGHER ORDER DERIVATIVES
Let y = f (x)
Example: Higher order

26. DIFFERENTIATION OF PARAMETRIC EQUATIONS
In mathematics, parametric equation is a method of
defining a curve using parameters. A simple example is
when one uses a time parameter to determine the
position, velocity, and other information about a body in
motion.
We define both x and y in terms of a third variable called
a parameter:

27. DIFFERENTIATION OF PARAMETRIC EQUATIONS
First derivatives
Second derivatives
Example: Parametric equations

28. DIFFERENTIATION OF PARAMETRIC EQUATIONS
Example: Parametric equations
Given
Therefore,
So,

29. LET’s TRY! (Page 123) Answer: A curve have a parametric equations and
Find: (a)
(b)
When t=1
Answer:
(a)
(b)

30. APPLICATIONS Rate of change
Rate of change is such problems to find the rate at which some quantity is changing.
Step of solving 1
Make a simple sketch (if appropriate) 2
Set up an equation relating all of the relevant quantities 3
Chain rule 4
Substitute all known values 5
Solve
Example: Rate of change

31. Answer: Example: Rate of change Page 148
At time t, the volume of water in a leaking tank is V cm3,
where:
Find the rate of water flows from the tank at t=6 seconds.
Answer: 31

32. APPLICATIONS L’ Hopital’s rules
L’ Hopital’s rule is a general method for using derivatives to find limits of indeterminate form.
Indeterminate form
L’ Hopital’s rules
If we have
Then by using L’ Hopital’s rule

33. Divide the indeterminate form to 4 cases
This is the basic form. Just
apply L’ Hopital’s rule to
find the limit.
Case 2
Rewriting function as a ratio
function to find the basic form
2. Apply L’ Hopital rule

34. Case 3
Combining the term to
find the basic form
2. Apply L’ Hopital rule
Case 4
Introducing
2. Apply ln to the limit
Find basic form for the right
hand side limit
4. Apply L’ Hopital rule

35. Example 1: L’ Hopital’s rule
Case 1 (0/0)

36. Example 2: L’ Hopital’s rule
Case 1 (0/0)

37. Example 3: L’ Hopital’s rule
Case 1 (∞/∞)

38. Example 3: L’ Hopital’s rule
Case 2 (0.∞) 1 2

39. Example 4: L’ Hopital’s rule
Case 3 (∞-∞) 1 2 3

40. Example 5: L’ Hopital’s rule
Case 4 (00, 1∞ or ∞0) 1 2