DifferentiationSafdar Alam

Table Of Contents

• 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 f ( x + h ) – f ( x ) h> 0 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

Chain Rules

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

Product Rule• Used for Multiplication• Product: (F*DS + S*DF)• (First * Derivative of Second + Second * Derivative of First)

• 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

Quotient Rule

Used for Division Quotient: (B*DT – T*DB) B₂ (Bottom * Derivative of Top – Top * Derivative of Bottom over Bottom

Squared)

Example F(x) = (5x + 1) x F’(x) = (x)(5) – (5x + 1)(1) F’(x) = (5x) – (5x

+ 1) x₂ x₂ F’(x) = ( -1 ) x₂

Practice Problem• F(x)= 2 , (4x + 1)₂

• F’(x)=

Work Page

Trig Functions

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

Practice Problems• Y= tanx sinx

• Y’=

Work Page

Implicit Differentiation

We use implicit, when we can’t solve explicitly for y in terms of x.

Example:

Y ₂ = 2y dy dx

Practice Problems

• F(x) = x₃ + y₃ = 15

• F’(x) =

Work Page

Logarithmic Differentiation

This applies to chain rules and properties of logs Rules of Log

Multiplication- AdditionDivision- SubtractionExponents- Multiplication

Some key functions to rememberln(1) = 0ln(e) = 1ln(x)x = xln(x)

Practice Problems

• Y= (3)x

• Dy/Dx=

Work Page

Exponential Diff.

F’(x) e(u)= e(u) (du/dx)-Copy the Function and take the derivative of the

angle

Examples:Y = e(5x)Y’= 5e(5x)

Y= e(sinx)Y’= e(sinx)*(-cosx)

Practice Problems

• F(x)= e(5x + 1)• F’(x) =

• F(x)= e(tanx)• F’(x)=

Work Page

Derivative of Natural Log

Y= ln(x)Y’(x)= 1/u * du/dx

ExamplesY= ln(5X)Y’= 5/5X = 1/X

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 – 5ydy/dx = 16x – 5y

5x + 3y₂

FRQ

• 1971 AB 1 - ln(x₂ - 4)

E. Find H’(7) 1 * (2x) 2x . (X₂ - 4) (X₂ - 4) 2(7) 14 . 14 ( (7)₂ - 4 ) (49 – 4 ) 45

Sources• 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