August 2009
Beginner
256 pages
6h 29m
English
Content preview from The Manga Guide to Calculus
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196 Chapter 6 Let’s Learn About Partial Dierentiation!
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
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