1

8.3 Partial Derivatives

222 :derivative Partial xyxx

f

22

2

2 :derivative partial Second xy

f

Ex.

222 If yxxyxf ,

xyy

f

xyx

f4 :derivative partial Mixed

2

Functions of Several Variables

Chapter 8Lecture 28

2

Partial Derivatives Partial Derivatives

x

y

y

x

Unceratinty Analysis:

Calculation of the area of a letter size paper with measured length and width

A = A' - A

= (x x)(y y) - xy

= xy yx xy

xy yx

2122

isThat

/

yxA y

A

x

A

xyA

2

221222122

2

m 150

m 1211010105

? m 50m 5m 10

:RMS) (Use ExampleFor

A

...yxxyA

yxA//

3

Partial Derivatives Partial Derivatives

• The partial derivative of f with respect to x is the derivative of f with respect to x, when all other variables are treated as constants.

• Similarly, the partial derivative of f with respect to y is the derivative of f with respect to y, when all other variables are treated as constants.

• The partial derivatives are written

, , and so on.f x f y

4

Ex. 2( , ) 3 lnf x y x y x y

6 lnf

xy yx

2 13

fx x

y y

Ex.

2

( , ) xy yg x y e

2

2 1 xy ygxy e

y

Partial Derivatives Partial Derivatives

5

Ex. 4 3( , , ) 2f x y z xy z xy

4 3 2f

y z yx

3 34 2f

xy z xy

4 23f

xy zz

Partial Derivatives Partial Derivatives

6

Geometric Interpretation of Partial Derivatives

Plane y = bP

z = f (x, y)

( , )a b

f

x

is the slope of the tangent line at the point

P(a,b, f (a,b)) along the slice through y = b.

-3 -2 -1 0 1 2 3-3-113

-10

-5

0

5

10

22 yxz Ex.

7

Ex.2 3 5( , ) lnf x y x y x x y

Second-Order Partial DerivativesSecond-Order Partial Derivatives

23 3

22 20

fy x

x

2 22 1

6f f

xyy x x y y

22

2 26

f xx y

y y

8

Notation for Partial Derivatives

means xf

fx

means yf

fy

xy

ff xy

2

means

yx

ff yx

2

means

9

Marginal Cost: Linear Model Suppose you own a company that makes two models of speakers, the Ultra Mini and the Big Stack. Your total monthly cost (in dollars) to make x Ultra Minis and y Big Stacks is given by

Example:

20

x

C

.402000010 yxyxC ,,

What is the significance C/x and C/y?

Solution:The cost is increasing at a rate of $20 per additional Ultra Mini (if productions of Big Stacks is held constant).

40

y

C The cost is increasing at a rate of $40 per additional Big Stack (if productions of Ultra Mini is held constant).

10

Marginal Cost: Interaction Model Another possibility for the cost function in the previous example is the interaction model

Example:

yx

C1020 a1. .

xyyxyxC 10402000010 .,,

b. What is the marginal cost of manufacturing Big Stacks at a production level of 100 Ultra Minis and 50 Big Stacks per month?

Solution:The marginal cost of manufacturing Ultra Minis increases by $0.1 for each Big Stack that is manufactured.

a. What are the marginal costs of the two models of speakers?

11

Stack. Bigper 501001040 Thus,

50. and

100 month,per Stacks Big 50 and Minis Ultra100

of level production aat ,1040 From b.

50100

$.

.

,

y

C

y

x

xy

C

xy

C1040 a2. .

The marginal cost of manufacturing Big Stack increases by $0.1 for each Ultra Minis that is manufactured.

12

Market Share (Cars and Light Trucks) Based on data from 1980-1998, the relationship between the domestic market shares of three major U.S. manufacturers of cars and light trucks is

Solution:

Exercise: (Waner, Problem #43, Section 8.3)

213 02022660 xxx ... where x1, x2, and x3 are, respectively, the fraction of the market held by Chrysler, Ford, and General Motors. Calculate x3/x1 and x1/x3. What do they signify, and how are they related to each other?

22 a1.1

3 .

x

x General Motors’ market share decreases by 2.2% per 1% increase in Chrysler’s market share if Ford’s share is unchanged.

13

22

1 a2.

3

1

.

x

x

Chrysler’s market share decreases by 1% per 2.2% increase in General Motors’ market share if Ford’s share is unchanged.

1

3

1

1

3 b.

x

x

x

x

That is, the two partial derivatives are reciprocals of each other.