Transcript 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.

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
w
,
x
w  x 2  xy  y2  2yz  2z 2  z,
w
w
, and
.
y
z
In order to find w x , we regard x as the
variable and y and z as constants.
w
 2x  y
x
Example 1 (cont.):
w  x  xy  y  2yz  2z  z,
2
and
2
w
y
 x  2y  2z
w
z

2y  4z  1
2
Example 2:
For
f (x, y)  3x y  xy, find fx and fy .
2
f x  6xy  y
Treating y as a constant
f y  3x  x
Treating x2 and x as a constants
2
Example 3:
For
f (x, y)  e  y ln x, find fx and fy .
xy
fx


fy
1
ye  y
x
y
xy
ye 
x
xy

x  e  1 ln x

xe  ln x
xy
xy
Example 4: A cellular phone company
has the following production function for a
23 13
certain product: p(x, y)  50x y ,
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.
Example 4 (cont.):
a) p(125,64)  50(125) (64)
2 3

13
5000 units
b) Marginal Productivity of Labor
p

 px
x
13
2 1 3 1 3 100y
 50  x y 
13
3
3x
 50(25)(4)
Example 4 (cont.):
Marginal Productivity of Capital
p

 py
y
2 3
1 2 3 2 3 50x
 50  x y 
2 3
3
3y
Example 4 (cont.):
c) Marginal Productivity of Labor
 px 125, 64 
100 64 
13

3125 
100  4

3 5
2
 26
3
13
Example 4 (cont.):
Marginal Productivity of Capital
 py 125, 64 
50 125 
2 3

364 
50  25

316
1
 26
24
2 3
DEFINITION:
Second-Order Partial Derivatives
Take the partial with
2 z
2 f
2 z 2 f
1.

 2  2  f xx
xx xx x
x
z
 f

 fxy
yx yx
2
2.
2
respect to x, and then
with respect to x again.
Take the partial with
respect to x, and then
with respect to y.
DEFINITION (cont.):
Take the partial with
2 z
2 f

 f yx
xy xy
3.
 z
 f
 z  f
4.

 2  2  f yy
yy yy y
y
2
2
2
2
respect to y, and then
with respect to x.
Take the partial with
respect to y, and then
with respect to y again.
Example 5: For
2 3
4
y
z  f (x, y)  x y  x y  xe ,
find the four second-order partial derivatives.
2 f
a) 2
x

fxx


 f
b)
yx
2

fxy



3
3
y
(2xy  4 x y  e )
x
2y 3  12x 2 y

(2xy 3  4 x 3 y  e y )
y
6xy 2  4 x 3  e y
Example 5 (cont.):
z  f (x, y)  x y  x y  xe
2 3
2 f
c)
xy

fyx


 f
d) 2
y
2

fyy


4
y

2 2
4
y
(3x y  x  xe )
x
2
3
y
6xy  4x  e

2 2
4
y
(3x y  x  xe )
y
2
y
6x y  xe