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Differentiation
Safdar Alam
Table Of Contents
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Chain Rules
Product/Quotient Rules
Trig
Implicit
Logarithmic/Exponential
Notations of Differentiation
• In functions you will see:
-f’(x)
-y’(x)
• These symbols are used to tell that the
function is a derivative
• Derivative: lim
h> 0
f(x+h)–f(x)
h
Formula for Derivative
• Nu↑(n-1)
• N, standing for a constant (which a derivative of a constant is zero)
• U, standing for a function
• Example: X₂
• Answer: 2x
Practice Problems
• F(x)= 5x₄
• F’(x)=
• F(x)= x₂+3x₂
• F’(x)=
Work Page
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Definition: Formula for the derivative of the two function
There are two types of chain rules. (Product/Quotient Rule)
Product: (F*DS + S*DF)
Quotient: (B*DT – T*DB)
B₂
Chain Rules
Product Rule
• Used for Multiplication
• Product: (F*DS + S*DF)
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(First * Derivative of Second + Second * Derivative of First)
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Example:
Y = (4x + 3)(5x)
Y’= (4x + 3)(5) + (5x)(4)
Y’= (20x + 15) + (20x)
Y’= 40x + 15
Practice Problem
• Y= (6x + 4)₂(25x + 13)
• Y’=
Work Page
Used for Division
 Quotient: (B*DT – T*DB)
B₂
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(Bottom * Derivative of Top – Top * Derivative of Bottom over Bottom
Squared)
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Example
F(x) = (5x + 1)
x
F’(x) = (x)(5) – (5x + 1)(1)
– (5x + 1)
x₂
x₂
F’(x) = ( -1 )
x₂
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Quotient Rule
F’(x) = (5x)
Practice Problem
• F(x)=
• F’(x)=
2 ,
(4x + 1)₂
Work Page
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Derivative of Trig. Functions
Sin(x) = Cos(x) dx
Cos(x) = -Sin(x) dx
Sec(x)= (secx)(tanx) dx
Tan(x)= Sec₂(x) dx
Csc(x)= -(cscx)(cotx) dx
Cot(x)= -csc₂(x) dx
Example:
Y= cos(x) + sin(x)
Y’= -sinx + cosx
Trig Functions
Practice Problems
• Y= tanx
sinx
• Y’=
Work Page
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We use implicit, when we can’t solve explicitly for y in
terms of x.
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Example:
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Y ₂ = 2y dy
dx
Implicit Differentiation
Practice Problems
• F(x) = x₃ + y₃ = 15
• F’(x) =
Work Page
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This applies to chain rules and properties of logs
Rules of Log
Multiplication- Addition
Division- Subtraction
Exponents- Multiplication
Some key functions to remember
ln(1) = 0
ln(e) = 1
ln(x)x = xln(x)
Logarithmic Differentiation
Practice Problems
• Y= (3)x
• Dy/Dx=
Work Page
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F’(x) e(u)= e(u) (du/dx)
-Copy the Function and take the derivative of the
angle
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Examples:
Y = e(5x)
Y’= 5e(5x)
Y= e(sinx)
Y’= e(sinx)*(-cosx)
Exponential Diff.
Practice Problems
• F(x)= e(5x + 1)
• F’(x) =
• F(x)= e(tanx)
• F’(x)=
Work Page
Y= ln(x)
 Y’(x)= 1/u * du/dx
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Examples
 Y= ln(5X)
 Y’= 5/5X = 1/X
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Derivative of Natural Log
Practice Problems
• Y= ln(ex)
• Y’=
• Y=ln(tanx)
Work Page
FRQ
• 1995 AB 3
-8x₂ + 5xy + y₃ = -149
A. Find dy/dx
-16x + 5x(dy/dx) + 5y + 3y₂(dy/dx) = 0
(dy/dx)(5x + 3y₂) = 16x – 5y
dy/dx = 16x – 5y
5x + 3y₂
FRQ
• 1971 AB 1
- ln(x₂ - 4)
E. Find H’(7)
1 * (2x)
(X₂ - 4)
2(7)
( (7)₂ - 4 )
2x .
(X₂ - 4)
14 .
(49 – 4 )
14
45
Sources
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http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/implicitdiffdirectory/ImplicitDiff.html
http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/logdiffdirectory/LogDiff.html
http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/chainruledirectory/ChainRule.html
© Safdar Alam
March 4, 2011