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8 Differentiation Rules F

Differentiation is the process of finding the derivative of a function, which represents the slope of the tangent line at a point and the instantaneous change at that point. The document outlines various differentiation rules for algebraic, transcendental, and trigonometric functions, including the limit definition of the derivative and the chain rule for composite functions. Additionally, it provides examples and formulas for differentiating different types of functions.

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seantinsay2006
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15 views36 pages

8 Differentiation Rules F

Differentiation is the process of finding the derivative of a function, which represents the slope of the tangent line at a point and the instantaneous change at that point. The document outlines various differentiation rules for algebraic, transcendental, and trigonometric functions, including the limit definition of the derivative and the chain rule for composite functions. Additionally, it provides examples and formulas for differentiating different types of functions.

Uploaded by

seantinsay2006
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Differentiation

Rules
What is differentiation?
• Differentiation is the process of finding the
derivative of a function.
What is derivative?
• The derivative of a function at a point is the slope
of the tangent line drawn to the curve at that point.
It also represents the instantaneous change at a
point on the function.
Limit Definition of Derivative

• The derivative of the function 𝑓, denoted by 𝑓′, given that


its value at a number 𝑥 in the domain of 𝑓 is given by
𝒇 𝒙+∆𝒙 −𝒇(𝒙)
𝒇′(𝒙) = 𝐥𝐢𝐦 ∆𝒙
∆𝒙→𝟎
if the limit exists.
Differentiation
Rules
(Algebraic Function)
Derivative of a Constant

d
(c) = 0 ( c is a constant )
dx
Derivative of a Power Function

dx
( )
d n
x = nx n −1
( n is a real number )
Derivative of a Constant Times a Function

d d
( cf ( x) ) = c ( f ( x) ) ( c is a constant )
dx dx
Derivative of Sum and Difference

d d d
 f ( x )  g ( x ) =  f ( x)   g ( x)
dx dx dx
Derivative of a Product

d d d
 f ( x )  g ( x ) =  f ( x) g ( x) +  g ( x) f ( x)
dx dx dx
Derivative of a Quotient

d d
d  f ( ) =
x  g ( x )
dx
 f ( x )  − f ( x )
dx
 g ( x ) 
 
dx  g ( x)   g ( x) 2
Sometimes remembered as:

d  hi  lo d  hi  − hi d lo
  =
dx  lo  lo lo
1. 𝑓 𝑥 =5
2. 𝑓 𝑥 =𝑥+ 𝑥 Differentiate the
3. 𝑓 𝑥 = 𝑥 5 − 3𝑥 following functions:
4. 𝑓 𝑥 = 3𝑥 7
5. 𝑓 𝑥 = −5𝑥
6. 𝑓 𝑥 = 5𝑥 3 − 4𝑥 2
7. 𝑓 𝑥 = 6𝑥 7 + 5𝑥 4 − 3𝑥 2 + 5
4
8. 𝑓 𝑥 =
𝑥6
9. 𝑓 𝑥 = 6𝑥 3 − 𝑥 10 − 20𝑥
𝑥+1
10.𝑓 𝑥 = 𝑥 2 −1
Differentiation
Rules
(Transcendental Function)
Derivative of Exponential
and Logarithmic Functions
Differentiate the following:

• 𝑓 𝑥 = 4𝑒 𝑥
•𝑓 𝑥 = 4 𝑥

• 𝑓′ 𝑥 = 4𝑒 𝑥 • 𝑓′ 𝑥 = 4 𝑥 ln 4
Differentiate the following:
• 𝑓 𝑥 = 2𝑥 log2 𝑥 • 𝑓 𝑥 = 2𝑥 ln 𝑥

1 1
• 𝑓′(𝑥) = 2𝑥 ln 2 log2 𝑥 + 2𝑥 𝑥 ln 2
• 𝑓′ 𝑥 = 2 1 ln 𝑥 + 𝑥 𝑥

• 𝑓′(𝑥) = 2𝑥 ln 2 log2 𝑥 + 𝑥 ln 2
1 • 𝑓′ 𝑥 = 2 ln 𝑥 + 1
• 𝑓′(𝑥) = 2 ln 𝑥 + 2
ln 2 log2 𝑥 𝑥 ln 2 +1
• 𝑓′(𝑥) = 2𝑥 𝑥 ln 2
Derivative of Trigonometric Functions
Differentiate 𝑓 𝑥 = 3 sin 𝑥 + 5 cos 𝑥.

𝑓′(𝑥) = 3(cos 𝑥) + 5 (− sin 𝑥)


𝑓′(𝑥) = 3 cos 𝑥 − 5 sin 𝑥
Differentiate 𝑓 𝑥 = sec 𝑥 + 3 csc 𝑥.

𝑓′ 𝑥 = sec 𝑥 𝑡𝑎𝑛 𝑥 + 3 − csc 𝑥 𝑐𝑜𝑡 𝑥


𝑓′ 𝑥 = sec 𝑥 𝑡𝑎𝑛 𝑥 −3 csc 𝑥 cot 𝑥
2
Differentiate 𝑓 𝑥 = 𝑥 s𝑖𝑛 𝑥 − 3𝑥 c𝑜𝑠 𝑥 + 5 sin 𝑥.

𝑓 ′ (𝑥) = [𝑥 2 cos 𝑥 + sin 𝑥 (2𝑥)] − 3[(𝑥)(− sin 𝑥) + cos 𝑥 (1)] + 5 cos 𝑥


𝑓′(𝑥) = 𝑥 2 cos 𝑥 + 2𝑥 sin 𝑥 + 3𝑥 sin 𝑥 − 3 cos 𝑥 + 5 cos 𝑥
𝑓 ′ 𝑥 = 2
𝑥 cos 𝑥 + 5𝑥 sin 𝑥 + 2 cos 𝑥
1. 𝑓 𝑥 = 2 𝑥 − cos 𝑥
Differentiate the
2. 𝑓 𝑥 = 10 − log 𝑥
sin 𝑥
following functions:
3. 𝑓 𝑥 = tan 𝑥 + 𝑥𝑒 𝑥
4. 𝑓 𝑥 = sec 𝑥 tan 𝑥 + cot 𝑥
5. 𝑓 𝑥 = 5 ln 𝑥 + 𝑥 2 log 𝑥
𝑥−1
6. 𝑓 𝑥 = ln 𝑥
Chain Rule of
Differentiation
Chain Rule of Differentiation

• Chain rule is a method of finding the derivative of


composite functions, or functions that are made by
combining one or more functions. It exists for
differentiating a function of another function.
Chain Rule
Another version of Chain Rule

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