196 Chapter 6 Let’s Learn About Partial Dierentiation!
In general,
∂
∂
=
∂
∂
=−
h
v
t
h
v
vt,.
98
Therefore, from w on page 194, near (v, t) = (v
0
, t
0
),
hvttvv vttt hv t
,.
,
()
≈−
()
+−
()
−
()
+
()
0000
00
0
98
Next, we’ll try imitating the concentration of sugar syrup given y
grams of sugar in x grams of water.
fxy
y
xy
f
x
f
y
xy
f
y
f
xy y
x
x
y
,
()
=
+
∂
∂
==−
+
()
∂
∂
==
+
()
−×
+
100
100
100 100 1
2
yy
x
xy
()
=
+
()
22
100
Thus, near (x, y) = (a, b), we have
fxy
b
ab
xa
a
ab
yb
b
ab
,
()
≈−
+
()
−
()
+
+
()
−
()
+
+
100 100 100
22
Definition of Partial Differentiation
When z = f(x, y) is partially differentiable with respect to x for every point
(x, y) in a region, the function (x, y)
→
f
x
(x, y), which relates (x, y) to f
x
(x, y),
the partial derivative at that point with respect to x, is called the partial dif-
ferential function of z = f(x, y) with respect to x and can be expressed by any
of the following:
ffxy
f
x
z
x
xx
,,,,
()
∂
∂
∂
∂
Similarly, when z = f(x, y) is partially differentiable with respect to y for
every point (x, y) in the region, the function
xy fxy
y
,,
()
→
()
Total Dierentials 197
is called the partial differential function of z = f(x, y) with respect to y and is
expressed by any of the following:
f f x y
f
y
z
y
y y
, , , ,
( )
∂
∂
∂
∂
Obtaining the partial derivatives of a function is called partially
differentiating it.
From the imitating linear function of z = f(x, y) at (x, y) = (a, b), we have
found
f x y f a b x a f a b x b f a b
x y
, , , ,
( )
≈
( )
−
( )
+
( )
−
( )
+
( )
We now modify this as
f x y f a b
f
x
a b x a
f
y
a b y b, , , ,
( )
−
( )
≈
∂
∂
( )
−
( )
+
∂
∂
( )
−
( )
Since f(x, y) − f(a, b) means the incre-
ment of z = f(x, y) when a point moves from
(a, b) to (x, y), we write this as Δz, as we
did for the single-variable functions.
Also, (x − a) is Δx and (y − b) is Δy.
Then, expression can be written as
{
∆ ≈
∂
∂
∆ +
∂
∂
∆z
z
x
x
z
y
y
This expression means, “If x increases from a by Δx and y from b by Δy
in z = f(x, y), z increases by
∂
∂
∆ +
∂
∂
∆
z
x
x
z
y
y
Total Differentials
2nd
Period
Total
dierentials
∂z
∂y
∆y
∆z
∆y
∆x
∂z
∂x
∆x
∂z
∂x
∆x
∂z
∂y
∆y
= +

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