9.2 Partial Derivatives

Find the partial derivatives of a given function.

Evaluate partial derivatives.

Find the four second-order partial derivatives of a function in two variables.

Example 1: For

find

In order to find , we regard x as the variable and y and z as constants.

w x2 xy y2 2yz 2z2 z,

w

x,

w

y, and

w

z.

w

x2x y

w x

Example 1 (cont.):

w

y x 2y 2z

andw

z 2y 4z 1

w x2 xy y2 2yz 2z2 z,

Example 2:

For f (x, y) 3x2y xy, find fx and fy .

fx 6xy y

fy 3x2 x

Treating y as a constant

Treating x2 and x as a constants

Example 3:For

f (x, y) exy y ln x, find fx and fy .

fx yexy y1

x

yexy y

x

fy x exy 1ln x

xexy ln x

Example 4: A cellular phone company

has the following production function for a

certain product:

where p is the number of units produced with

x units of labor and y units of capital.

a) Find the number of units produced with

125 units of labor and 64 units of capital.

b) Find the marginal productivities.

c) Evaluate the marginal productivities at

x = 125 and y = 64.

p(x, y) 50x2 3y1 3,

Example 4 (cont.):

a) p(125,64) 50(125)2 3(64)1 3 50(25)(4)

5000 units

b) Marginal Productivity of Labor

p

xpx

502

3x 1 3y1 3

100y1 3

3x1 3

Example 4 (cont.):

Marginal Productivity of Capital

p

ypy

501

3x2 3y 2 3

50x2 3

3y2 3

Example 4 (cont.):

c) Marginal Productivity of Labor

px 125,64

100 64 1 3

3 125 1 3

1004

35

262

3

Example 4 (cont.):

Marginal Productivity of Capital

py 125,64

50 125 2 3

3 64 2 3

5025

316

261

24

DEFINITION:

Second-Order Partial Derivatives

1.

2z

xx

2 f

xx

2z

x2 2 f

x2 fxx

Take the partial with

respect to x, and then

with respect to x again.

2.2z

yx

2 f

yx fxy

Take the partial with

respect to x, and then

with respect to y.

DEFINITION (cont.):

3.2z

xy

2 f

xy fyx

Take the partial with

respect to y, and then

with respect to x.

4.2z

yy

2 f

yy

2z

y2 2 f

y2 fyy

Take the partial with

respect to y, and then

with respect to y again.

Example 5: For

find the four second-order partial derivatives.

z f (x, y) x2y3 x4 y xey ,

a) 2 f

x2 fxx x

(2xy3 4x3y ey )

2y3 12x2y

b) 2 f

yx fxy

y

(2xy3 4x3y ey )

6xy2 4x3 ey

Example 5 (cont.):

c) 2 f

xy fyx

x

(3x2y2 x4 xey )

6xy2 4x3 ey

d) 2 f

y2 fyy y

(3x2y2 x4 xey )

6x2y xey

z f (x, y) x2y3 x4 y xey