Differentiation

3• Basic Rules of Differentiation

• The Product and Quotient Rules

• The Chain Rule

• Marginal Functions in Economics

• Higher-Order Derivatives

Basic Differentiation Rules

1.

Ex.

2.

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex.

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3.

Ex.

4.

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex.

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules5.

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

( ) ( ) ( ) ( ) d d d

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

Product Rule

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 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules6.

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as:

More Differentiation Rules6.

Ex.2

3 5( )

2

xf x

x

Quotient Rule (cont.)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7. The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note: h(x) is a composite function.

If ( ) , where ( ), theny h x g u u f x

dy dy du

dx du dx

Another Version:

More Differentiation Rules

The General Power Rule:

If ( ) ( ) , real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex. 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

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 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex.

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

Marginal Functions

The Marginal Cost Function approximates the change in the actual cost of producing an additional unit.

The Marginal Average Cost Function measures the rate of change of the average cost function with respect to the number of units produced.

The Marginal Profit Function measures the rate of change of the profit function. It approximates the profit from the sale of an additional unit.

The Marginal Revenue Function measures the rate of change of the revenue function. It approximates the revenue from the sale of an additional unit.

ExampleThe monthly demand for T-shirts is given by

0.05 25 0 400p x x

where p denotes the wholesale unit price in dollars and x denotes the quantity demanded. The monthly cost function for these T-shirts is

2( ) 0.001 2 200 C x x x

1. Find the revenue and profit functions.

2. Find the marginal cost, marginal revenue, and marginal profit functions.

Solution

20.05 25 0.05 25 x x x x

2 20.05 25 0.001 2 200x x x x

1. Find the revenue and profit functions.

2. Find the marginal cost, marginal revenue, and marginal profit functions.

Revenue = xp

Profit = revenue – cost

20.049 23 200x x

Marginal Cost = ( )C x

0.002 2 . . . . . .x

Solution2. (cont.) Find the marginal revenue and marginal profit functions.

Marginal revenue = ( )R x0.1 25x

0.098 23x ( )P xMarginal profit =

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative.

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives5 3( ) 3 2 14f x x x Given find ( ).f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2).f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f