Matematika Dasar

PEM Akamigas

1
Chapter 3
Derivative

2
 Tangent Line
• Tangent Line is a straight line that “just touch” a point in a curve
• The point of searching tangent line is define a rate of change of
a function in the interval as small as possible
• For example, a line goes through point and called secant line.
The slope of those secant line is:

• To get tangent line, we have to get secant line with the smallest
interval possible, then the interval must be close to zero

3
 Example 1

Solve
Find slope of tangent line at point
Answer

4
 Example 2

Solve
Find slope of tangent line at point (2,4)
Answer

5
 Instantaneous Velocity
• We calculate average velocity as

• To get instantaneous velocity at certain time , we have to get

6
 Example 3

Solve
An object moves at position function . What is the instantaneous velocity at
Answer

7
 Example 3

Solve
An object moves at position function cm. What is the instantaneous velocity
at
Answer

cm/s

8
 Derivative
• The derivative of a function is
•
• If the limit is exist, then the function is differentiable at
• Differentiability is also show that the function is continuous at x,

because the curve cannot take a jump

9
 Increments
• If the value of changes from to , then the change in - is called
increment of and commonly written as
• For example, in the picture there is gap between and (0.4), then

• If then

10
 Leibniz Notation of Derivative
• Suppose the variable changes from to , then the variable y also
changes to

The ratio of will be

As then the slope of secant line touches the tangent line

11
Graph of Derivative

Tangent line has

Tangent line has

12
 Rules of Finding Derivative(1)

Proof

Proof

13
 Rules of Finding Derivative(2)

Proof:

Proof

14
 Rules of Finding Derivative(3)

Proof

15
 Example 4

Solve

Answer

16
 Rules of Finding Derivative(4)

Proof

17
 Example 5
Solve

Answer

18
 Derivative of Trigonometric
Functions(1)

Proof

19
 Derivative of Trigonometric
Functions(2)

Proof

20
 Example 6
Solve

Answer

21
 Example 7
Solve

Answer

22
 The Chain Rule
• To differentiate composite function, we can use chain rule, which consist of:

or

23
 Example 8
Solve

Answer

Let

24
 Example 9
Solve

Answer

Let

25
 Example 10
Solve

Answer

Let

26
 Higher Order Derivative
• We can differentiate a function multiple times, for example:

then

27
 Example 11(1)
Solve
We throw ball upward from a building 192 ft and initial velocity of 64 ft/s. The ball follow the
trajectory of a function
• When does it reach maximum height?
Answer

The ball reach maximum height when

28
 Example 11(2)
Solve
We throw ball upward from a building 192 ft and initial velocity of 64 ft/s. The ball follow the
trajectory of a function
• What is its maximum height?
• When does the ball hit the ground
Answer

When

29
 Example 11(3)
Solve
We throw ball upward from a building 192 ft and initial velocity of 64 ft/s. The ball follow the
trajectory of a function
• What the velocity of the ball when hit the ground?
• What is its acceleration at
Answer

For

30
 Implicit Differentiation
We can use chain rule to differentiate implicit function, such as

For example

Let

31
 Example 12
Prove

Answer

We will use the chain rule

32
 Example 13
Solve

Answer

33
 Example 14
Solve

Answer

Let

34
Related Rates
If the function depends on time, then its derivative is called time rates of change
Example
A balloon is released 150 m from the observer, who is on the level ground. The
balloon goes straight up with at a rate of 8 m/s. How fast the rate of distance from
observer to balloon increasing when balloon is 50 m height?
Answer
The relation between and is defined by

Differentiate both sides with chain rule and we got

35
Related Rates

when

36
Example 15
Water flow through the conical tank at the rate of . The height of tank is 12 m and the
radius of the circular opening is 6 m. How fast the water level rising when the water is 4 m
deep? (hint : find the relation between and for this tank first!)
Answer

37
 Inverse Derivative
If

Example
If
Answer

38
 Logarithmic Derivative

Proof:

39
 Example 15
Solve

Answer

Let

40
 Logarithmic Derivative

Proof: Proof:

41
 Example 16
Solve

Answer

Let
Let

42
 Example 17
Solve

Answer

43
Continuity of Function
• If is exists, then continuous at c
• But the converse is not true, If continuous at c, is not necessarily exists
• For example
• is a continuous function, but it is not differentiable at
Proof :

At

At

44
 Continuity of Function

There are several function that is not differentiable at certain point c, such as:
• Stepwise function, function that is not continuous
• Function that has a sharp cusp
• Function that has vertical tangent line

45