Techniques of Differentiation

• The Product and Quotient Rules

• The Chain Rule

• Derivatives of Logarithmic and Exponential Functions

• Implicit Differentiation

The Quotient Rule

2

( ) ( ) ( ) ( )

( ) ( )

f xd f x g x f x g x

dx g x g x

( ) ( ) ( ) ( ) d

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

The Product Rule

Ex. 3 7 2( ) 2 5 3 8 1f x x x x 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

Derivative of first

Derivative of Second

The Product Rule

The Quotient Rule

Ex.2

3 5( )

2

xf x

x

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

Compute the Derivative

Ex. 3 2

13 2

x

dx x x x

dx

2 2 3

11 9 2 3 2 2

xx x x x x x

1 9 1 2 1 3 2 2

= –10

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.

Ex. 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.

Ex. 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 Experiment

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

1n nd duu n u

dx dx

Ex. 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

The Chain Rule7

2 1( )

3 5

xG 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

Ex.

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 Example5 2 8 2, 7 3y u u x x Ex.

dy dy du

dx du dx

3 2 7556 6

2u 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

Differentiation of 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

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

1( ) 2f x

x

Ex. 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:

Differentiation of Logarithmic Functions

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

Differentiation of Logarithmic Functions

4log 2 3 4d

x xdx

Ex.

4 4log 2 log 3 4d

x xdx

1 1( 4)

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

Derivative of Logarithms of Absolute Values

1log

lnbd du

udx u b dx

1ln

d duu

dx u dx

Derivative of Logarithms of Absolute Values

Ex. 2ln 8 3d

xdx

2

116

8 3x

x

Ex. 31

log 2d

dx x

2

1 1

1/ 2 ln 3x x

2

1

2 ln 3x x

Differentiation of Exponential Functions

x xde e

dx

u ud due e

dx dx

Generalized Rule for eu:

Derivative of ex:

If u is a differentiable function, then

Derivatives of Exponential Functions

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

3 4 4 4( ) 4 4x xf x x e x e

3 44 1xx e x

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

3 5( ) 3 5x df x e x

dx

3 55 xe

Differentiation of Exponential Functions

lnx xdb b b

dx

lnu ud dub b b

dx dx

Generalized Rule for bu:

Derivative of bx:

If u is a differentiable function, then

Derivatives of Exponential Functions

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

2 27 2 2x x x

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

dx

Implicit Differentiation

33 4 17y x x

y is explicitly a function of x.

3 3 1y xy x

y is implicitly a function of x.

Implicit Differentiation (cont.)

23 3dy dy

y y xdx dx

23 3dy

y x ydx

2

3

3

dy y

dx y x

Solve for

To differentiate the implicit case we use the chain rule where y is a function of x:

dy

dx

Tangent Line to Implicit Curve

Ex. Find the equation of the tangent line to the curve at the point (2, 1).3 ln

2

xy y

2 1 13

2

dy dyy

dx y dx

13 1

2

dy dy

dx dx

1

8

dy

dx

11 2

8y x

Logarithmic DifferentiationEx. Use logarithmic differentiation to find the derivative of 5

3 2 9 1 .y x x

5ln ln 3 2 9 1y x x

1ln ln 3 2 5ln 9 1

2y x x

1 3 5(9)

2 3 2 9 1

dy

y dx x x

5 3 45

3 2 9 12 3 2 9 1

dyx x

dx x x

Apply ln

Differentiate

Properties of ln

Solve