Lecture Note Function of Two Variables
Treating y as a constant, we obtain
83
z
x y
x∂
= −
∂
Treating x as a constant, we obtain
310
z
x y
y∂
=− +
∂
Example 2
Find the partial derivatives fxand fyif ()^22
2
,2
3
y
fxy x xy
x=+ +.
Example 3
For the functionf(xy xe, )=+xy y^2 , findffxy(1, 2 a n d ) (1, 2).
Solution
() () () (),^2
0
(1 )
xy xy
x
xy xy
xyf xy e x x e y
xxx
exye
exy∂ ∂∂
=++
∂∂∂
=+ +
=+
Now we evaluatefx()1, 2 , fx()1,2=+×=ee^12 × (112 3)^2
fy()xy x xe,2= ×+= +xy y xe^2 xy 2 y
Now we evaluatefy()1, 2 , fey()1, 2 = 1212 ×+×=+× 2 2 e^24
Example 4
Suppose that the production function Qxy( ,2000)= x y0.5 0.5^ is known. Determine the
marginal productivity of labor and the marginal productivity of capital when 16 units
of labor and 144 units of capital are used.
Solution
()()
0.5
0.5 0.5
0.5
0.5
0.5 0.5
0.51000
2000 0.5
1000
2000 0.5
Qy
xy
xx
Qx
xy
yy−−∂
==
∂
∂
==
∂
Substitutingxy==16 and^144 , we obtain
0.5
0.5
(16,144)
1000(144) 1000 12
3000 units
(1 6 ) 4Q
x∂×
===
∂
and
0.5
0.5
(16,144)1000(16) 1000 4
333.33 units
(144) 12Q
y∂×
===
∂
Thus we see that adding one unit of labor will increase production by about 3000
units and adding one unit of capital will increase production by about 333 units.
Example 5
It is estimated that the weekly output at a certain plant is given by the function
Qxy(), =++−−1, 200x 500 y xy x y^232 units, where x is the number of skilled