Chapter 4Techniques of Differentiation

Sections 4.1, 4.2, and 4.3

Techniques of Differentiation

The Product and Quotient Rules

The Chain Rule

Derivatives of Logarithmic and Exponential asFunctions

1 12) or n n n ndx nx x nx

dx

1) 0 a constant or 0d

c c cdx

3) ( ) ( ) or ( ) ( )d d

cf x c f x cf x cf xdx dx

4) ( ) ( ) d

f x g x f x g xdx

Available Rules for Derivatives

2

( ) ( ) ( ) ( )6)

( ) ( )

f xd f x g x f x g x

dx g x g x

5) ( ) ( ) ( ) ( ) d

f x g x f x g x f x g xdx

Two More Rules

The product rule

The quotient rule

If f (x) and g (x) are differentiable functions, then we have

3 7 2If ( ) 2 5 3 8 1 , find ( )f x x x x x f x

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 230 48 105 40 45 80 2x x x x x x

The Product Rule - Example

Derivative of first

Derivative of Second

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of numerator

Derivative of denominator

The Quotient Rule - Example

2

3 5If ( ) , find ( )

2

xf x f x

x

Calculation Thought Experiment

Given an expression, consider the steps you would use in computing its value. If the last operation is multiplication, treat the expression as a product; if the last operation is division, treat the expression as a quotient; and so on.

Example:

2 4 3 6x x

To compute a value, first you would evaluate the parentheses then multiply the results, so this can be treated as a product.

2 4 3 6 5x x x

To compute a value, the last operation would be to subtract, so this can be treated as a difference.

Calculation Thought ExperimentExample:

The Chain Rule

( ) ( )d du

f u f udx dx

The derivative of a f (quantity) is the derivative of f evaluated at the quantity, times the derivative of the quantity.

If f is a differentiable function of u and u is a differentiable function of x, then the composite f (u) is a differentiable function of x, and

Generalized Power Rule

17) n nd duu n u

dx dx

Example: 1 22 23 4 3 4d d

x x x xdx dx

1 221

3 4 6 42

x x x

2

3 2

3 4

x

x x

72 1

If ( ) find ( )3 5

xG x G x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 137

3 5 3 5 3 5

xx

x x x

Generalized Power Rule

Example:

Chain Rule in Differential Notation

If y is a differentiable function of u and u is a differentiable function of x, then

dy dy du

dx du dx

Chain Rule Example

5 2 8 2If and 7 3 , find dy

y u u x x ydx

dy dy du

dx du dx 3 2 75

56 62

u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Logarithmic Functions

1ln 0

dx x

dx x

1ln

d duu

dx u dx

Generalized Rule for Natural Logarithm Functions

Derivative of the Natural Logarithm

If u is a differentiable function, then

Find the derivative of 2( ) ln 2 1 .f x x

1( ) 2f x

x

Find an equation of the tangent line to the graph of ( ) 2 ln at 1, 2 .f x x x

2

2

2 1( )

2 1

dx

dxf xx

2

4

2 1

x

x

(1) 3f

2 3( 1)

3 1

y x

y x

Slope: Equation:

Examples

1log

lnbd

xdx x b

1log

lnbd du

udx u b dx

Generalized Rule for Logarithm Functions.

Derivative of a Logarithmic Function.

If u is a differentiable function, then

More Logarithmic Functions

4log 2 3 4d

x xdx

4 4log 2 log 3 4d

x xdx

1 1( 4)

( 2) ln 4 (3 4 ) ln 4x x

Examples

Logarithms of Absolute Values

1log

lnbd du

udx u b dx

1ln

d duu

dx u dx

2ln 8 3d

xdx

2

116

8 3x

x

31

log 2d

dx x

2

1 1

1/ 2 ln 3x x

2

1

2 ln 3x x

Examples

Exponential Functions

x xde e

dx

u ud due e

dx dx

Generalized Rule for the natural exponential function.

Derivative of the natural exponential function.

If u is a differentiable function, then

Find the derivative of 3 5( ) .xf x e

4 4( ) xf x x e

3 44 1xx e x

Find the derivative of 4 4( ) xf x x e

3 5( ) 3 5x df x e x

dx 3 55 xe

3 4 4 44 4x xx e x e

4 4 4 4x xx e x e

Examples

lnx xdb b b

dx

lnu ud dub b b

dx dx

Generalized Rule for general exponential functions.

Derivative of general exponential functions.

If u is a differentiable function, then

Exponential Functions

Find the derivative of 2 2( ) 7x xf x

2 22 2 7 ln 7x xx

2 2 2( ) 7 ln 7 2x x df x x x

dx

Exponential Functions