Basic

Calculus

Derivative
(Lesson 3)
March 14, 2024
Mrs. Irene Oriarte
* Rules in Computer Lab
* Rules in using the
tablet/cellular phone
*Attendance Check
 The Basic
Differentiation Rules
 Where do
you usually
see this?

4
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.
5
 This speed is called the
instantaneous rate of change, or
simply the instantaneous speed.
This concept is related to limits and
derivatives.

6
What are you feeling
at this point in time?
 How are you feeling?

A B C

D E F
 • Apply the differentiation rules in computing
the derivative of an algebraic, exponential,
and trigonometric functions (STEM_BC11D-
IIIf-3).
Objectives • Identify the different real-life application of
derivatives
• Relate derivatives in another field.
• Work as a team in identifying the derivatives of
a function
ASSIGNMENT
Try It!

Give an example of real-life

Exponent
Exponent of x of the
of x Derivative

application
Assignmentof Calculus or
●

Derivatives
ASSIGNMENT
Try It!

Exponent
Exponent of x of the

What have you learned from

of x Derivative

Assignment
●
the video?
ASSIGNMENT
Try It!

Exponent
Exponent of x of the

“Seeing the world in another

of x Derivative

Assignment
●
language “
Try It!
“Seeing the world in another language “

-Can you identify “other language” where Exponent

you can
Exponent of x of the
relate calculus as the mathematics of change?
of x Derivative

● Assignment
Examples: in education, socio economic, culture,
etc.

If there is a concern, how do we address them?

 • Apply the differentiation rules in computing
the derivative of an algebraic, exponential,
and trigonometric functions (STEM_BC11D-
IIIf-3).
Objectives • Identify the different real-life application of
derivatives
• Relate derivatives in another field.
• Work as a team in identifying the derivatives of
a function
ASSIGNMENT
Try It!

Exponent
Exponent of x of the
of x Derivative

● Assignment
Names:
Try It! ___________________________________

Exponent
Exponent of x of the
of x Derivative

● Assignment
Try It!
ASSIGNMENT
Try It!

Exponent
Exponent of x of the
of x Derivative

● Assignment
0 0 0 0
0 0 0 0
1 1 1 0
2 2x 2 1
3 3x2 3 2
 The Basic Differentiation Rules

Constant Rule
The derivative of a constant function is
0. If 𝑐 is a constant, then

𝒅
𝒄 = 𝟎.
𝒅𝒙
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 The Basic Differentiation Rules

Example:
a.If 𝑓 𝑥 = 6, then 𝑓 𝑥 = 0.
′

b.Given than 𝑦 = 24, 𝑦 = 0.

′

21
 The Basic Differentiation Rules

Power Rule
If 𝑘 is a real number, then the
derivative of the function 𝑓 𝑥 = 𝑥 is
𝑘

𝒅 𝒌 𝒌−𝟏
𝒙 = 𝒌𝒙 .
𝒅𝒙
22
 The Basic Differentiation Rules

Example:
Find the derivative of 𝑓 𝑥 = 𝑥 .
4

23
 The Basic Differentiation Rules

Example:
Find the derivative of 𝑓 𝑥 = 𝑥 .
4

The derivative of 𝑓 𝑥 = 𝑥 is 𝑓 𝑥 4 ′

= 4𝑥 .
3

24
 The Basic Differentiation Rules

Constant Multiple Rule

If 𝑓 is differentiable and 𝑐 is a constant,
then the derivative of the function 𝑐 ∙ 𝑓 is

𝒅 ′
𝒄𝒇 𝒙 = 𝒄𝒇 𝒙 .
𝒅𝒙

25
 The Basic Differentiation Rules

Example:
Use the Constant Multiple Rule to find
the derivative of 𝑓 𝑥 = 4𝑥 .
6

26
 The Basic Differentiation Rules

Example:
Use the Constant Multiple Rule to find
the derivative of 𝑓 𝑥 = 4𝑥 .
6

The derivative of 𝑓 𝑥 = 4𝑥 6 is
𝑓 𝑥 = 24𝑥 .
′ 5
27
 The Basic Differentiation Rules

Example:
Use the Constant Multiple Rule to find
the derivative of 𝑓 𝑥 = 3𝑥.

28
 The Basic Differentiation Rules

Example:
Use the Constant Multiple Rule to find
the derivative of 𝑓 𝑥 = 3𝑥.

The derivative of 𝑓 𝑥 = 3𝑥 is
′
𝑓 𝑥 = 3.
29
 The Basic Differentiation Rules

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 ′ ′
𝒇 𝒙 +𝒈 𝒙 =𝒇 𝒙 +𝒈 𝒙 .
𝒅𝒙 30
 The Basic Differentiation Rules

Sum or Difference Rule

The derivative of the sum of 𝑓 and 𝑔 is
given by

𝒅 ′ ′
𝒇 𝒙 −𝒈 𝒙 =𝒇 𝒙 −𝒈 𝒙 .
𝒅𝒙

31
 The Basic Differentiation Rules

Example:
Use the Sum or Difference Rule to
find the derivative of
𝑓 𝑥 =𝑥 +𝑥 .
3 2

32
 The Basic Differentiation Rules

Example:
Use the Sum or Difference Rule to find the
derivative of 𝑓 𝑥 = 𝑥 + 𝑥 .
3 2

The derivative of the function 𝑓 𝑥 = 𝑥 3 + 𝑥 2 is

𝑓 ′ 𝑥 = 3𝑥 2 + 2𝑥.

33
What differentiation
rules can be used to get
the derivative of
𝟑 𝟐
𝒇 𝒙 = 𝒙 + 𝟑𝒙 + 𝟓?

34
What differentiation
rules can be used to get
the derivative of
−𝟑
𝒇 𝒙 = ?
𝒙𝟐
35
Tips

In differentiating functions of
the form 𝒇 𝒙 = 𝒄 ∙ 𝒙 , we can find
𝒌

the derivative of the function by

multiplying the exponent to the
coefficient of the function and
subtracting 1 from the exponent.

36
Tips

To differentiate the function

𝒇 𝒙 = 𝟑𝒙 , we can directly
𝟔

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 .
37
Remember

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.
38
 Check Your Understanding

Find the derivative of each function.

1. 𝑓 𝑥 = 26
2. 𝑦 = 4 − 2𝑥
3. 𝑔 𝑥 = −9𝑥 4

1
4. 𝑓 𝑥 = 3
𝑥
5. 𝑦 = 2
3𝑥 + 𝑥+1
39
 Check Your Understanding

Find the derivative of each function.

1. 𝑓 𝑥 = 26
2. 𝑦 = 4 − 2𝑥
3. 𝑔 𝑥 = −9𝑥 4

1
4. 𝑓 𝑥 = 3
𝑥
5. 𝑦 = 2
3𝑥 + 𝑥+1 6. All are easy
40
What have you learned from
the lesson today?
The End
Check Your Understanding