DERIVATIVE

𝒇 𝒙 = 𝐥𝐢𝐦∆𝒙→𝟎

𝒇 𝒙+ ∆𝒙 −𝒇(𝒙)

∆𝒙 =

𝒅𝒚

𝒅𝒙 = y’

if the limit exists.

For y = f(x), the derivative of f at x, denoted by f(x), to be

𝑑𝑦

𝑑𝑥

“derivative of y with respect to x” or “dee y over dee x”

- means that the rate of change of y is based on the change on the value of x.

TAKE NOTE:

𝒅𝒚

𝒅𝒙 is NOT beinng regarded as quotient, but as a

single symbol.

Four-Step Differentiation Process 1. Replace x by 𝒙 + ∆𝒙 and y by

𝒚 + ∆𝒚.

2. Solve for ∆𝒚 in terms of 𝒙 + ∆𝒙.

3. Divide both sides by ∆𝒙.

4. Find the limit of ∆𝑦

∆𝑥 as ∆𝒙 0.

Examples

1.y = 2x – 3

2. y = 4x - 12

3. y = x - 2

Increment Method 1.Evaluate 𝑓 𝑥 + ℎ .

2.Subtract by 𝑓 𝑥 .

3.Divide by h.

4.Find the limit as h 0

Examples

1. y = 2x – 3

2. y = 4x - 12

3. y = x - 2

Rules in finding the DERIVATIVES

The Constant Function Rule

If y = f(x) = C, where C is a constant, then y’ = 0.

Also, 𝑑𝑦

𝑑𝑥 = 0 and f’(x) = 0

Rules in finding the DERIVATIVES

The Identity Function Rule

If y = f(x) = x, where x is a differentiable function,

then y’ = 1.

Also, 𝑑𝑦

𝑑𝑥 = 1 and f’(x) = 1

Rules in finding the DERIVATIVES

The Constant Multiple Rule

If y = f(x) = 𝐶 ∗ 𝑓(𝑥), where f(x)

is a differentiable function, then y’ = 𝐶 ∗ 𝑓′(𝑥)

Also, 𝑑𝑥

𝑑𝑦 = 𝐶 ∗ 𝑓′(𝑥) and f’(x) =

𝐶 ∗ 𝑓′ 𝑥 .

Rules in finding the DERIVATIVES

The Sum and Difference Rule

If y = f(x) = 𝑢 𝑥 ± 𝑣(𝑥)where

u and v are differentiable functions, then y’ = u′(x) ± v’(x)

Also, 𝑑𝑦

𝑑𝑥 =

u′ x ± 𝑣′ 𝑥 𝑎𝑛𝑑 𝑢′ 𝑥 =𝑢′ 𝑥 ± 𝑣′(𝑥)

Rules in finding the DERIVATIVES

The Power Rule

If y = f(x) = 𝑥𝑛, where x is a

differentiable function and n is a real number, then y’ = 𝑛𝑥𝑛−1.

Also, 𝑑𝑦

𝑑𝑥 = 𝑛𝑥𝑛−1 and f’(x) =

𝑛𝑥𝑛−1

FLASH IT! (INDIVIDUAL TASK)

Some students find it hard to memorize the different rules in differentiation. In this performance task, you are to make at least 5 flash cards involving differentiation rules. In this flash card, you need to put in all rules in differentiation (Sum rule, Constant Multiple rule, etc.) Take note that, in front of your flash card you must state the rule for differentiation and on its back, write at least 3 examples. You will be graded according to content, creativity and punctuality. Put your flash cards in an envelope or anything that will keep your flash cards together.

Anchor Good(5) Adequate(3) Poor(1) Weight Score

Content

Cards contain rules for differentiation and examples were correct.

Cards contain rules for differentiations were some examples are not correct.

Cards contain rules for differentiations were examples are not correct.

5 25

Creativity

The cards are presentable and were colourful and neat.

The cards are presentable and somewhat colourful and neat.

The cards are not presentable and not colourful and neat.

3 15

Compliance

There were atleast 5 cards. and were colorful and neat

There were less than 5 cards and were somewhat presented creatively.

The fan did not comply with the size.

3 15

Date of submission

The cards were submitted on the day of submission

The cards were submitted a day after the date of submission

The cards were submitted 2 days after the date of submission

2 10

Total 70

Rules in finding the DERIVATIVES

The Product Rule

If y = f(x) = u(x) * v(x), where u and v are differentiable functions,

then y’ = 𝒖 𝒙 ∗ 𝒗′(𝒙) + v(x) * u’(x).

Also, 𝒅𝒚

𝒅𝒙 = 𝒖 𝒙 ∗ 𝒗′ 𝒙 + 𝒗 𝒙 ∗

𝒖′(𝒙) and f’(x) = 𝒖 𝒙 ∗ 𝒗′ 𝒙 +𝒗 𝒙 ∗ 𝒖′(𝒙)

Rules in finding the DERIVATIVES The Quotient Rule

If y = f(x) = 𝒖(𝒙)

𝒗(𝒙), where u and v

are differentiable function then,

y’ = 𝒗 𝒙 ∗𝒖′ 𝒙 −𝒖 𝒙 ∗𝒗′(𝒙)

[𝒗 𝒙 ]𝟐

Also, f’(x) = 𝒗∗𝒖′−𝒖∗𝒗′

𝒗𝟐 and

𝒅𝒚

𝒅𝒙=

𝒗∗𝒅𝒖

𝒅𝒙−𝒖∗

𝒅𝒗

𝒅𝒙

𝒗𝟐

SEATWORK #1

I. Find the derivatives of the following functions

A. By applying the rules

1. 𝒚 = 𝒙−𝟐𝟎

2. 𝒚 =𝟑

𝟒𝒙𝟓

3. y = 𝟐−𝟓𝒙

𝟑𝒙−𝟏

4. 𝒚 = −𝟓𝒙(𝒙𝟒 − 𝟒)

5. y = 3x

6. y = - 𝟏𝟖

7. y = 𝟔𝒙𝟑 − 𝟏𝟐𝒙𝟐 + 𝟕

8. y = 𝒙 −𝟐

𝒙𝟑

B. using the four-process differentiation

1. y = 𝟓𝒙 −10

C. Using the increment method

2. y = 18x + 2