Applied Mathematics for Business and Economics

Lecture Note Function of Two Variables

workers and y the number of unskilled workers employed at the plant. Currently the
work force consists of 30 skilled workers, and 60 unskilled workers. Use maginal
analysis to estimate the change in the weekly output that will reslt from the addition
of 1 more skilled worker if the number of unskilled workers is not changed.

(Answer: x(30, 60)=2,100

)

Q units.)

2.2 Second-Order Partial Derivatives

Ifzfxy= (, , the partial derivative of fxwith respect to x is:

(^) ()
2
xx x x 2
zz
ffor
x xx

∂ ∂∂⎛⎞

==⎜⎟

∂ ∂∂⎝⎠

The partial derivative of fx with respect to y is

()

2

xy x y

zz

ffor

y xyx

∂ ∂∂⎛⎞

==⎜⎟

∂∂∂∂⎝⎠

The partial derivative of fy with respect to x is

(^) ()
2
yx y or
x
zz
ff
x yxy

∂ ∂∂⎛⎞

==⎜⎟

∂∂∂∂⎝⎠

The partial derivative of fy with respect to y is

2

yy y yor 2

zz

ff

yyy

∂ ∂∂⎛⎞

==⎜⎟

∂ ∂∂⎝⎠

Example 6
Compute the four second-order partial derivatives of the function

f()x y,52=+ ++xy^32 xy x 1.

Solution

Sincefy yx=+ +^3252 , it follows that fxx= 0 andfxy=+ 310 y^2 y. Since

3102

fy=+xy xy, it follows that

3102

fyx=+yyand fyy= 6 xy+ 10 x.

Example 7

Find all four second partial derivatives off(xy,ln 4)=+(x^2 y), then findfxx()2,1 2.

Solution
We must find the first partial derivatives fxand fybefore we can find the second

partial derivatives.

2

2

2

2

12

2

44

14

4

44

x

y

x

fx

x yx

f

y

x yx

=×=

++

=×=

++y

()

() ()

(^22)
2222
(^422228)
44
xx
xy xxx y
f
x yx

+×−× −+

==

++y