Chapter 8

Multivariable Calculus

Section 2

Partial Derivatives

2

Learning Objectives for Section 8.2 Partial Derivatives

The student will be able to evaluate partial derivatives and second-order partial derivatives.

3

Introduction to Partial Derivatives

We have studied extensively differentiation of functions of one variable. For instance, if

C(x) = 100 + 500x,

then

C ´(x) = 500.

If we have more than one independent variable, we can still differentiate the function if we consider one of the variables independent and the others fixed. This is called a partial derivative.

4

Example

For a company producing only one type of surfboard, the cost function is C(x) = 500 + 70x, where x is the number of boards produced.

Differentiating with respect to x, we obtain the marginal cost function C ´(x) = 70.

Since the marginal cost is constant, $70 is the change in cost for a one-unit increase in production at any output level.

5

Example(continued)

For a company producing two types of boards, a standard model and a competition model, the cost function is

C(x, y) = 700 + 70x + 100y,

where x is the number of standard boards, and y is the number of competition boards produced.

Now suppose that we differentiate with respect to x, holding y fixed, and denote this by Cx(x, y); or suppose we differentiate with respect to y, holding x fixed, and denote this by Cy(x, y).

Differentiating in this way, we obtain Cx(x, y) = 70, and Cy(x, y)= 100.

6

Example(continued)

Each of these is called a partial derivative, and in this example, each represents marginal cost.

The first is the change in cost due to a one-unit increase in production of the standard board with the production of the competition board held fixed.

The second is the change in cost due to a one-unit increase in production of the competition board with the production of the standard board held fixed.

7

Partial Derivatives

If z = f (x, y), then the partial derivative of f with respect to x is defined by

h

yxfyhxf

x

zh

),(),(lim

0

and is denoted by ),(oror yxffx

zxx

The partial derivative of f with respect to y is defined by

k

yxfkyxf

y

zk

),(),(lim

0

and is denoted by ),(oror yxffy

zyy

8

Example

Let f (x, y) = 3x2 + 2xy – y 3 .

a. Find f x(x, y). [This is the derivative with respect to x, so consider y as a constant.]

9

Example

Let f (x, y) = 3x2 + 2xy – y 3 .

a. Find f x(x, y). [This is the derivative with respect to x, so consider y as a constant.]

f x(x, y) = 6x + 2y

b. Find f y(x, y). [This is the derivative with respect to y, so consider x as a constant.]

10

Example

Let f (x, y) = 3x2 + 2xy – y 3 .

a. Find f x(x, y). [This is the derivative with respect to x, so consider y as a constant.]

f x(x, y) = 6x + 2y

b. Find f y(x, y). [This is the derivative with respect to y, so consider x as a constant.]

f y(x, y) = 2x – 3y2

c. Find f x(2,5).

11

Example

Let f (x, y) = 3x2 + 2xy – y 3 .

a. Find f x(x, y). [This is the derivative with respect to x, so consider y as a constant.]

f x(x, y) = 6x + 2y

b. Find f y(x, y). [This is the derivative with respect to y, so consider x as a constant.]

f y(x, y) = 2x - 3y2

c. Find f x(2,5).

f x(2,5) = 6 · 2 + 2 · 5 = 22.

12

Example Using the Chain Rule

Let f (x, y) = (5 + 2xy 2) 3.

Hint: Think of the problem as z = u3 and u = 5 + 2xy2.

a. Find f x(x, y)

13

Example Using the Chain Rule

Let f (x, y) = (5 + 2xy 2) 3.

Hint: Think of the problem as z = u3 and u = 5 + 2xy2.

a. Find f x(x, y)

f x(x, y) = 3 (5 + 2xy2)2 · 2y2

b. Find f y(x, y)

The chain.

14

Example Using the Chain Rule

Let f (x, y) = (5 + 2xy 2) 3.

Hint: Think of the problem as z = u3 and u = 5 + 2xy2.

a. Find f x(x, y)

f x(x, y) = 3 (5 + 2xy2)2 · 2y2

b. Find f y(x, y)

f y(x, y)) = 3 (5 + 2xy2)2 · 4xy

The chain.

The chain.

15

Second-Order Partial Derivatives

Taking a second-order partial derivative means taking a partial derivative of the first partial derivative. If z = f (x, y), then

x

z

yxy

z (x, y) f f yxyx

2

y

z

xyx

z (x, y) f f xyxy

2

y

z

yy

z (x, y) f f yyyy 2

2

x

z

xx

z (x, y) f f xxxx 2

2

16

Example

Let f (x, y) = x3 y3 + x + y2.

a. Find f x(x, y).

17

Example

Let f (x, y) = x3 y3 + x + y2.

a. Find f xx(x, y).

f x(x, y) = 3x2 y3 + 1

f xx(x, y) = 6xy3

b. Find f xy(x, y).

18

Example

Let f (x, y) = x3 y3 + x + y2.

a. Find f x(x, y).

f x(x, y) = 3x2 y3 + 1

f xx(x, y) = 6xy3

b. Find f xy(x, y).

f x(x, y) = 3x2 y3 + 1

f xy(x, y) = 9x2 y2

19

Summary

■ If z = f (x, y), then the partial derivative of f with respect to x is defined by

■ If z = f (x, y), then the partial derivative of f with respect to y is defined by

■ We learned how to take second partial derivatives.

h

yxfyhxf

x

zh

),(),(lim

0

k

yxfkyxf

y

zk

),(),(lim

0