Applied Mathematics for Business and Economics

Lecture Note Function of Two Variables

()

() ()

()

() ()

()

()()

2

2222

2

2222

2

2222

(^40248)
44
(^40428)
44
(^404416)
44
xy
yx
yy
xy x x
f
x yx
xy x
y
x
f
x yx
xy
f
xy xy

+×−× −

==

++

+×−× −

==

++

+×−× −

==

++

y

()

2

22

2

1

22 8

182 4

2,

(^23142)
24
2
fxx

−× +⎛⎞

⎜⎟ −+ −

⎛⎞===⎝⎠

⎜⎟

⎝⎠⎛⎞+

⎜⎟+×

⎝⎠

41

69

=−

Example 8
Suppose the output Q at a factory depends on the amount K of capital invested in the
plant and equipment and also on the size L of the labor force, measured in worker-
hours. Give an economic interpretation of the sign of the second-order partial

derivative

2

2

Q

L

∂

∂

.

Solution

If

2

2

Q

L

∂

∂

is negative, the marginal product of labor

Q

L

∂

∂

decreases as L increases. This

implies that for a fixed level of capital investment, the effect on output of the addition
of 1worker-hour of labor is greater when the work force is small than when the work
force is large.

Similarly, if

2

2

Q

L

∂

∂

is positive, it follows that for a fixed level of capital investment, the

effect on output of the addition of 1 worker-hour of labor is greater when the work
force is larger than when it is small.

Remark The two partial derivatives fxyand fyxare sometimes called the mixed

second-order partial derivatives of f andfxyy=fx.

3 The Chain Rule; Approximation by the Total Differential

3.1 Chain Rule for Partial Derivatives
Recall that if z is a differentiable function of x and x is a differentiable function of t,
then z can be regarded as a differentiable function of t and the rate of change of z with
respect to t is given by the chain rule
dz dz dx
dt dx dt

=

Here is the corresponding rule for functions of two variables.