Math 200, Spring 2010
Handout 11
Section 14-3
Partial Derivatives
Partial Derivatives
The
partial derivatives
are the rates of change with respect to each variable separately.
A function
( )
,
f x y
in two variables has two partial derivatives, denoted
x
f
and
y
f
, defined by the following limits (if they
exist):
( )
( ) ( )
0
,
,
,
lim
x
h
f a
h b
f a b
f
a b
h
→
+
−
=
, which is the
partial derivative of
f
with respect to
x
at
( )
,
a b
and
( )
( ) ( )
0
,
,
,
lim
y
h
f a b
h
f a b
f
a b
h
→
+
−
=
, which is the
partial derivative of
f
with respect to
y
at
( )
,
a b
Thus,
( )
,
x
f
a b
is the obtained by keeping
y
fixed
( )
y
b
=
and finding the
ordinary
derivative at
a
of the
function
( ) ( )
,
g x
f x b
=
, and
( )
,
y
f
a b
is the obtained by keeping
x
fixed
( )
x
a
=
and finding the
ordinary
derivative at
b
of the function
( ) ( )
,
G y
f a y
=
.
Rule for Finding Partial Derivatives of
z
=
f
(
x
,
y
)
1.
To find
x
f
, regard
y
as a constant and differentiate
( )
,
f x y
with respect to
x
.
2.
To find
y
f
, regard
x
as a constant and differentiate
( )
,
f x y
with respect to
y
.
1.
Find the first partial derivatives of
the function
( )
4
3
2
,
8
f x y
x y
x y
=
+
.
2.
Find the partial derivative
( )
2,1
x
f
for the function
( )
4
3
2
,
8
f x y
x y
x y
=
+
.
"otations for Partial Derivatives