Differentiation
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Transcript of Differentiation
DERIVATIVE
๐ ๐ = ๐ฅ๐ข๐ฆโ๐โ๐
๐ ๐+ โ๐ โ๐(๐)
โ๐ =
๐ ๐
๐ ๐ = yโ
if the limit exists.
For y = f(x), the derivative of f at x, denoted by f(x), to be
๐๐ฆ
๐๐ฅ
โderivative of y with respect to xโ or โdee y over dee xโ
- means that the rate of change of y is based on the change on the value of x.
TAKE NOTE:
๐ ๐
๐ ๐ is NOT beinng regarded as quotient, but as a
single symbol.
Four-Step Differentiation Process 1. Replace x by ๐ + โ๐ and y by
๐ + โ๐.
2. Solve for โ๐ in terms of ๐ + โ๐.
3. Divide both sides by โ๐.
4. Find the limit of โ๐ฆ
โ๐ฅ as โ๐ 0.
Examples
1.y = 2x โ 3
2. y = 4x - 12
3. y = x - 2
Increment Method 1.Evaluate ๐ ๐ฅ + โ .
2.Subtract by ๐ ๐ฅ .
3.Divide by h.
4.Find the limit as h 0
Examples
1. y = 2x โ 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 yโ = 0.
Also, ๐๐ฆ
๐๐ฅ = 0 and fโ(x) = 0
Rules in finding the DERIVATIVES
The Identity Function Rule
If y = f(x) = x, where x is a differentiable function,
then yโ = 1.
Also, ๐๐ฆ
๐๐ฅ = 1 and fโ(x) = 1
Rules in finding the DERIVATIVES
The Constant Multiple Rule
If y = f(x) = ๐ถ โ ๐(๐ฅ), where f(x)
is a differentiable function, then yโ = ๐ถ โ ๐โฒ(๐ฅ)
Also, ๐๐ฅ
๐๐ฆ = ๐ถ โ ๐โฒ(๐ฅ) and fโ(x) =
๐ถ โ ๐โฒ ๐ฅ .
Rules in finding the DERIVATIVES
The Sum and Difference Rule
If y = f(x) = ๐ข ๐ฅ ยฑ ๐ฃ(๐ฅ)where
u and v are differentiable functions, then yโ = uโฒ(x) ยฑ vโ(x)
Also, ๐๐ฆ
๐๐ฅ =
uโฒ x ยฑ ๐ฃโฒ ๐ฅ ๐๐๐ ๐ขโฒ ๐ฅ =๐ขโฒ ๐ฅ ยฑ ๐ฃโฒ(๐ฅ)
Rules in finding the DERIVATIVES
The Power Rule
If y = f(x) = ๐ฅ๐, where x is a
differentiable function and n is a real number, then yโ = ๐๐ฅ๐โ1.
Also, ๐๐ฆ
๐๐ฅ = ๐๐ฅ๐โ1 and fโ(x) =
๐๐ฅ๐โ1
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 yโ = ๐ ๐ โ ๐โฒ(๐) + v(x) * uโ(x).
Also, ๐ ๐
๐ ๐ = ๐ ๐ โ ๐โฒ ๐ + ๐ ๐ โ
๐โฒ(๐) and fโ(x) = ๐ ๐ โ ๐โฒ ๐ +๐ ๐ โ ๐โฒ(๐)
Rules in finding the DERIVATIVES The Quotient Rule
If y = f(x) = ๐(๐)
๐(๐), where u and v
are differentiable function then,
yโ = ๐ ๐ โ๐โฒ ๐ โ๐ ๐ โ๐โฒ(๐)
[๐ ๐ ]๐
Also, fโ(x) = ๐โ๐โฒโ๐โ๐โฒ
๐๐ and
๐ ๐
๐ ๐=
๐โ๐ ๐
๐ ๐โ๐โ
๐ ๐
๐ ๐
๐๐
SEATWORK #1
I. Find the derivatives of the following functions
A. By applying the rules
1. ๐ = ๐โ๐๐
2. ๐ =๐
๐๐๐
3. y = ๐โ๐๐
๐๐โ๐
4. ๐ = โ๐๐(๐๐ โ ๐)
5. y = 3x
6. y = - ๐๐
7. y = ๐๐๐ โ ๐๐๐๐ + ๐
8. y = ๐ โ๐
๐๐
B. using the four-process differentiation
1. y = ๐๐ โ10
C. Using the increment method
2. y = 18x + 2