Applied Mathematics for Business and Economics

Lecture Note Function of Two Variables

Suppose z is a function of x and y, each of which is a function of t then z can

be regarded as a function of t and

dz z dx z dy

dt x dt y dt

∂ ∂

=+

∂∂

Remark 1
zdx
xdt

∂

=

∂

rate of change of z with respect to t for fixed y.

zdy

ydt

∂

=

∂

rate of change of z with respect to t for fixed x.

Example 1

Find if^22 3 , 2 1, and

dz

zx xyx t yt

dt

=+ =+ =.

Solution
By the chain rule,

()23 232

dz z dx z dy

x yx

dt x dt y dt

∂∂

=+=+×+×

∂∂

t

Which you can rewrite in terms of t by substituting x= 21 t+ andyt=^2 to get

4(21)6 3(21)(2)18 14 4^22

dz

tttttt

dt

=++++ =++

Example 2
A health store carries two kinds of multiple vitamins, Brand A and Brand B. Sales
figures indicate that if Brand A is sold for x dollars per bottle and Brand B for y
dollars per bottle, the demand for Brand A will be

Qxy(), =− +300 20x^230 ybottles per month

It is estimated that t months from now the price of Brand A will be
xt=+2 0.05 dollars per bottle

and the price of Brand B will be

yt=+2 0.1 dollars per bottle

At what rate will the demand for Brand A be changing with respect to time 4 months
from now?

Solution

Your goal is to find

dQ

dt

when t= 4. Using the chain rule, you get

(^40) ()0.05 30 0.05()^12
dQ Q dx Q dy
dt x dt y dt
xt−

∂ ∂

=+

∂∂

=− +

whentx==+×=4, 2 0.05 4 2.2

and hence,

40 2.2 0.05 30 0.05 0.5 3.65

dQ

dt

=− × × + × × =−