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.