2. DERIVATIVE
For y = f(x), the derivative of f at x, denoted by f(x),
to be
𝒇 𝒙+ ∆𝒙 −𝒇(𝒙) 𝒅𝒚
𝒇 𝒙 = 𝐥𝐢𝐦 = = y’
∆𝒙→𝟎 ∆𝒙 𝒅𝒙
if the limit exists.

3. 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
is NOT beinng regarded as quotient, but as a
𝒅𝒙
single symbol.

4. 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.
∆𝑦
4. Find the limit of as ∆𝒙  0.
∆𝑥

5. Examples
1.y = 2x – 3

6. 2. y = 4x - 12

7. 3. y = x - 2

8. Increment Method
1.Evaluate 𝑓 𝑥 + ℎ .
2.Subtract by 𝑓 𝑥 .
3.Divide by h.
4.Find the limit as h  0

9. Examples
1. y = 2x – 3

10. 2. y = 4x - 12

11. 3. y = x - 2

12. Rules in finding the DERIVATIVES
The Constant Function Rule
If y = f(x) = C, where C is
a constant, then y’ = 0.
𝑑𝑦
Also, = 0 and f’(x) = 0
𝑑𝑥

13. Rules in finding the DERIVATIVES
The Identity Function Rule
If y = f(x) = x, where x is a
differentiable function,
then y’ = 1.
𝑑𝑦
Also, = 1 and f’(x) = 1
𝑑𝑥

14. 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)
𝑑𝑥
Also, = 𝐶 ∗ 𝑓′(𝑥) and f’(x) =
𝑑𝑦
′
𝐶∗ 𝑓 𝑥 .

15. 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)
𝑑𝑦
Also, =
𝑑𝑥
′ ′ ′
u x ± 𝑣 𝑥 𝑎𝑛𝑑 𝑢 𝑥 =
𝑢 ′ 𝑥 ± 𝑣′(𝑥)

16. Rules in finding the DERIVATIVES
The Power Rule
𝑛
If y = f(x) = 𝑥 , where x is a
differentiable function and n is
a real number, then y’ = 𝑛𝑥 𝑛−1 .
𝑑𝑦 𝑛−1
Also, = 𝑛𝑥 and f’(x) =
𝑑𝑥
𝑛−1
𝑛𝑥

17. 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.

18. Anchor Good(5) Adequate(3) Poor(1) Weight Score
Cards contain
Cards contain Cards contain
rules for
rules for rules for
differentiations
Content differentiation differentiations 5 25
were some
and examples were examples
examples are not
were correct. are not correct.
correct.
The cards are
The cards are The cards are not
presentable and
presentable and presentable and
Creativity somewhat 3 15
were colourful not colourful and
colourful and
and neat. neat.
neat.
There were less
There were
than 5 cards and The fan did not
atleast 5 cards.
Compliance were somewhat comply with the 3 15
and were colorful
presented size.
and neat
creatively.
The cards were The cards were The cards were
Date of submitted on the submitted a day submitted 2 days
2 10
submission day of after the date of after the date of
submission submission submission
Total 70

19. 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).
𝒅𝒚 ′
Also, = 𝒖 𝒙 ∗ 𝒗 𝒙 + 𝒗 𝒙 ∗
𝒅𝒙
′
𝒖′(𝒙) and f’(x) = 𝒖 𝒙 ∗ 𝒗 𝒙 +
𝒗 𝒙 ∗ 𝒖′(𝒙)

20. Rules in finding the DERIVATIVES
The Quotient Rule
𝒖(𝒙)
If y = f(x) = , where u and v
𝒗(𝒙)
are differentiable function then,
𝒗 𝒙 ∗𝒖′ 𝒙 −𝒖 𝒙 ∗𝒗′ (𝒙)
y’ =
[𝒗 𝒙 ] 𝟐
𝒗∗𝒖′ −𝒖∗𝒗′
Also, f’(x) = 𝟐 and
𝒗
𝒅𝒖 𝒅𝒗
𝒅𝒚 𝒗∗ −𝒖∗
𝒅𝒙 𝒅𝒙
=
𝒅𝒙 𝒗𝟐

21. SEATWORK #1
I. Find the derivatives of the following
functions
A. By applying the rules
−𝟐𝟎
1. 𝒚 = 𝒙
𝟑 𝟓
2. 𝒚= 𝒙
𝟒
𝟐−𝟓𝒙
3. y=
𝟑𝒙−𝟏

22. 𝟒
4. 𝒚 = −𝟓𝒙(𝒙 − 𝟒)
5. y = 3x
6. y = - 𝟏𝟖
𝟑 𝟐
7. y = 𝟔𝒙 − 𝟏𝟐𝒙 + 𝟕
𝒙 −𝟐
8. y =
𝒙𝟑
B. using the four-process differentiation
1. y = 𝟓𝒙 −10

23. C. Using the increment method
2. y = 18x + 2