14.3: PARTIAL DERIVATIVES
KIAM HEONG KWA
Let
f
be a function of two variables whose domain
D
includes points
arbitrarily close to (
a, b
). The partial derivative of
f
with respect to
x
is given by
f
x
(
a, b
) = lim
h
→
0
f
(
a
+
h, b
)

f
(
a, b
)
h
provided the limit exists. Note that we keep the variable
y
fixed in the
limiting process. Likewise, the partial derivative of
f
with respect to
y
is given by
f
y
(
a, b
) = lim
h
→
0
f
(
a, b
+
h
)

f
(
a, b
)
h
provided the limit exists, where the variable
x
is kept fixed in taking
the limit. The functions
f
x
(
x, y
) = lim
h
→
0
f
(
x
+
h, y
)

f
(
x, y
)
h
and
f
y
(
x, y
) = lim
h
→
0
f
(
x, y
+
h
)

f
(
x, y
)
h
are referred to as the partial derivatives of
f
.
Remark 1.
Note that by fixing the variable
y
, say
y
=
b
, one identifies
the plane
y
=
b
from a multitude of planes parallel to the
xz
plane. The
intersection of this plane with the graph of
f
, i.e., the surface defined
by
z
=
f
(
x, y
)
, defines a curve that passes through the point
(
a, b
)
and
is given by the relation
z
=
f
(
x, b
)
(at least locally). In vector notation,
this curve is given by
r
(
x
) =
x
i
+
b
j
+
f
(
x, b
)
k
.
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 Fall '10
 Kwa
 Calculus, Geometry, Derivative, YZ

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