7.2 Partial Derivatives
The rate of change of a function with respect to one of its variables when the other is held constant.
Partial
Derivatives Suppose
.
The partial derivative of f with respect to x is denoted by
The partial derivative of f with respect to y is denoted by
If 
,
find 
If 
,
find
If 
,
find 
If 
,
find
Transparency – Page 544
Marginal
Analysis
#46 A
manufacturer estimates that the annual output at a certain factory is given by 
units,
where K is the capital expenditure in units of $1,000 and L is
the size of the labor force in worker-hours.
a)
Find the
marginal productivity of capital 
and the
marginal productivity of labor 
when the
capital expenditure is $630,000 and the labor level is 830 worker-hours.
b) Should the manufacturer consider adding a unit of capital or a unit of labor in order to increase output more rapidly?
Second-Order Partial Derivatives
If ,
The partial
derivative of 
with respect
to x is
The partial
derivative of with respect
to y is
The partial
derivative of 
with respect
to x is
The partial
derivative of with respect
to y is
Examples
Find second-order partial derivatives for:
Find second-order partial derivatives for:
