Transcript Section 13.3

Section 13.3
Partial Derivatives
To find f x you consider y constant and
differentiate with respect to x.
Similarly, to find f y you hold x
constant and differentiate with respect
to y.
Examples:
f ( x, y )  x  3 y  7
xy
z 2
2
x y
2
2
Geometrically speaking, the partial
derivatives of a function of two
variables represent the slopes of
the surfaces in the x- and ydirections.
No matter how many variables are
involved, partial derivatives can be
thought of as rates of change.
Example: Consider the Cobb-Douglas production
function f ( x, y)  200 x 0.7 y 0.3
When x=1000
and y=500, find
a. The marginal productivity of labor
b. The marginal productivity of capital.
Assume x represents labor, and y capital.
As we can do with functions of a
single variable, it is possible to take
second, third, and higher partial
derivatives
Notation:
  f 
2 f
 f xx


2
x  x 
x

y

y
 f

 y

 f

 x
2

 f

  y 2  f yy

2
 f

 f xy

yx

Examples:
1) Find the first four second partial derivatives.
f ( x, y)  2 xe  3 ye
y
x