kalkulus pertemuan ke 4 tentang derivative.pptx

CHAPTER 2. DERIVATIVE
Materials :
 Initial Concept of Derivative
 Gradient of Function at a Point
 Operation on Derivative
 Derivative Mind Map
Indicators of Achievement :
1. Students are able to explain the concept of derivative correctly.
2. Students are able to determine the average velocity, instantaneous
velocity, and gradient of a function at a point using the concept of
derivative.
3. Students are able to calculate the derivative and explain the operations
that apply to the derivative of a function.
9/26/23
Informatics Enginnering | Calculus 1C | Rima Aulia Rahayu, S.Si,. M.Si. 1
SUB-CPMK020301 | Pertemuan 4

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Tangent Line
Tangent line of the curve 𝑦 = 𝑓(𝑥) at point 𝑃(𝑥, 𝑓 𝑋 ) is a line that goes through 𝑃
with slope
𝑚 = lim
ℎ→0
𝑓 𝑥 + Δ𝑥 − 𝑓(𝑥)
Δ𝑥
= 2.5
2.1 Two Problems One Solution

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Average speed vs instantaneous speed
Suppose 𝑟(𝑡) is the position of an object at time 𝑡, hence average velocity at time
t1 to t2 be :
𝑣𝑎𝑣𝑔 𝑡1, 𝑡2 =
𝑟 𝑡2 − 𝑟(𝑡1)
𝑡2 − 𝑡1
The smaller the difference between 𝑡1 and 𝑡2, we can get the "instantaneous
speed":
𝑣 𝑡 = lim
Δ𝑡→0
𝑣𝑎𝑣𝑔 𝑡, 𝑡 + Δ𝑡 = lim
Δ𝑡→0
𝑟 𝑡 + Δ𝑡 − 𝑟(𝑡)
Δ𝑡

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Definition of Derivatives
A function 𝑓 is said to have a derivative at point 𝑥 if,
lim
Δ𝑥→0
𝑓 𝑥 + Δ𝑥 − 𝑓(𝑥)
Δ𝑥
There is and is up.
If the above limit exists, then the derivative of f at point x i.e. f′(x) is equal
to the limit value above. The form of the limit above is equivalent to,
𝑓′ 𝑐 = lim
x→𝑐
𝑓 𝑥 − 𝑓(𝑐)
x − c
Here are some notations for derivatives of f at point x,
𝑓′
𝑥 𝐷𝑥𝑓(𝑥)
𝑑
𝑑𝑥
𝑓(𝑥)
𝑑(𝑓 𝑥 )
𝑑𝑥
2.2 Derivative

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If f has a derivative at point x=c, then f is continuous at point x=c.
If f is not continuous at point x=c, then f has no derivative at point x=c.

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Constant Function Rules
If 𝑓 𝑥 = 𝑘, where k is a constant, then for any x,𝑓′
𝑥 = 0.
Identity function rules
If f x = 𝑥, maka 𝑓′
𝑥 = 1
Rank Rules
If 𝑓 𝑥 = 𝑥𝑛
, with n rational numbers, then𝑓′
𝑥 = 𝑛𝑥𝑛−1
Constant Multiplier Rules
𝑘𝑓 ′ 𝑥 = 𝑘 ∙ 𝑓′ 𝑥 , with k a constant
Addition and subtraction rules
𝑓 ∙ 𝑔 ′ 𝑥 = 𝑓 𝑥 𝑔′ 𝑥 + 𝑓′ 𝑥 𝑔 𝑥
Division Rules
𝑓
𝑔
′
𝑥 =
𝑔 𝑥 𝑓′
𝑥 − 𝑓 𝑥 𝑔′(𝑥)
𝑔2(𝑥)
, 𝑔′
𝑥 ≠ 0
2.3 Rules for searching derivative

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Trigonometric Derivatives
The sin x and cos x functions are differentiated across real numbers with
𝐷𝑥 sin 𝑥 = cos 𝑥
𝐷𝑥 cos 𝑥 = − sin 𝑥
Advanced Trigonometric Derivatives
The derivative of the trigonometric function below can be obtained from the derivative
rules in the previous chapter and utilizes the basic derivative of trigonometry
𝐷𝑥 tan 𝑥 = sec2
𝑥
𝐷𝑥 cot 𝑥 = − csc2
𝑥
𝐷𝑥(sec 𝑥) = sec 𝑥 tan 𝑥
𝐷𝑥 csc 𝑥 = − csc 𝑥 cot 𝑥
2.4 Derivatives of Trigonometric Functions

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Example, 𝑦 = 𝑓 𝑢 and 𝑢 = 𝑔 𝑥 .
𝑓 ∘ 𝑔 ′ 𝑐 = 𝑓′ 𝑔 𝑐 ∙ 𝑔′ 𝑐
Or
𝑑𝑦
𝑑𝑥
=
𝑑𝑦
𝑑𝑢
∙
𝑑𝑢
𝑑𝑥
Example:
Asked to look for derivatives of 𝑥2 + 3𝑥 3
𝑦 = 𝑢3, 𝑢 = 𝑥2 + 3𝑥
So
𝑑𝑦
𝑑𝑥
= 3 𝑥2
+ 3𝑥 2
∙ 2𝑥
2.5 Chain Rules

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Derivatives of Order 2 and 3
The second and third derivatives of the function f at point x are respectively
written 𝑓′′
𝑥 and 𝑓′′′
(𝑥) where 𝑓′′
(𝑥) is obtained by deriving 𝑓′
(𝑥) once and
𝑓′′′
(𝑥) is obtained by decreasing 𝑓′′
(𝑥) once again.
Derivatives of Order More than 3
The nth derivative of the function f at point x is written in the form 𝑓 𝑛
(𝑥).
2.6 Higher Order derivatives

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Implicit derivative y with respect to x.
For example, given the function of two variables f, g, and a curve equation
𝑓 𝑥, 𝑦 = 𝑔 𝑥, 𝑦
Can we determine the derivative of y to x?
We can proceed by considering y as a function of x (in other words, 𝑦(𝑥)), and
deriving both segments of this equation to x. These are referred to as implicit
derivatives.
2.7 Implicit Derivatives

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Example, Suppose we are asked to specify
𝑑𝑦
𝑑𝑥
from the following equation:
𝑥𝑦 = 𝑥2
+ 𝑦2
𝑑
𝑑𝑥
𝑥𝑦 =
𝑑
𝑑𝑥
𝑥2
+ 𝑦2
We apply the multiplication rule to the left field, and the addition rule to the right field
𝑑
𝑑𝑥
𝑥 ∙ 𝑦 + 𝑥 ∙
𝑑
𝑑𝑥
𝑦 =
𝑑
𝑑𝑥
𝑥2 +
𝑑
𝑑𝑥
𝑦2
Since we think of y as a function of x, we need to apply the chain rule to
𝑑
𝑑𝑥
𝑦2 .
1 ∙ 𝑦 + 𝑥 ∙
𝑑𝑦
𝑑𝑥
= 2𝑥 + 2𝑦 ∙
𝑑𝑦
𝑑𝑥
𝑥 ∙
𝑑𝑦
𝑑𝑥
− 2𝑦 ∙
𝑑𝑦
𝑑𝑥
= 2𝑥 − 𝑦
𝑑𝑦
𝑑𝑥
=
2𝑥 − 𝑦
𝑥 − 2𝑦

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Rate-related problem-solving strategies,
In general, the steps that can be done:
1. Create simple modeling
2. Determine variables and the relationship between variables
3. Calculate the implicit derivative
4. Search according to what is asked
2.8 Related Rate

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Approximation
Approximation with differentials is carried out by approaching Δ𝑦 with
𝑑𝑦 = 𝑓′ 𝑥 𝑑𝑥
Δy ≈ 𝑑𝑦
𝑓 𝑥 + Δ𝑥 − 𝑓(𝑥) ≈ 𝑓′ 𝑥 𝑑𝑥
𝑓 𝑥 + Δ𝑥 ≈ 𝑓 𝑥 + 𝑓′
𝑥 Δx
2.9 Approximation

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Example: Use differential to approximate 4.5 !
Suppose 𝑓 𝑥 = 𝑥 . We are asked to
approach 4.5 = 𝑓 4 + 0.5
𝑓 4 + 0.5 ≈ 𝑓 4 + 𝑓′(4) ∙ 0.5
Then we need to find out first.𝑓′
4 .
𝑓 𝑥 = 𝑥 = 𝑥
1
2 ⟹ 𝑓′
𝑥 =
1
2 𝑥
𝑓′
4 =
1
2.2
=
1
4
So 4.5 ≈ 2 +
1
4
∙ 0.5 = 2.215
We can use this technique to approach the
roots of other things. For example 5 with
𝑑𝑥 = 1 . But of course, the approximate
results will be less accurate as the 𝑑𝑥 gets
bigger.

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Specify the turnan of the following functions:
1. 2𝑥3 + 3𝑥
2. 4𝑥3
+ 𝑥 − 5 𝑥2
+ 2𝑥
3.
𝑥2+1
2𝑥−1

kalkulus pertemuan ke 4 tentang derivative.pptx

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