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