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Partial Derivatives
Consider a two variable function
, suppose we let only
vary while keeping
fixed, say
, where
is a constant. Then we are really considering a function of a single variable
, namely
. If
has a derivative at
, then we call it the partial derivative of
with respect to
at (
,
) and denote it by
(or
).
Thus by the definition of a derivative we have:
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Similarly, the partial derivative of
with respect to
at (
,
),
(or
) is given by:
![]()
Let's consider the function
, the graphs below show the geometric meaning of
and
:
Here is an alternate geometric interpretation for the partial derivative in terms of vector functions:
Let
and the graph
of the function
(viewed on the plane
.) is the curve traced out by the vector function G (
)=(
) whose vector derivative G '(
)=(
) is determined by
. (That is, its "slope" is
.)
similarly, the graph of
is the curve
traced out by the vector function H (
)=(
) whose vector derivative H '(
)=(
) is determined by
.
Now let's take a look at a function
which is not continuous at (
,
), but
exists at (
). Let
for all (
,
) except at (
,
), and
.
As we can see from the graphs above,
along
,
and
for all (
) except at (
) along
. So that
is not continuous at (
,
) (
does not exist), but
,
if (
) = (
).
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Note that
and
for all (
) other than (
,
).
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Now let's consider the partial derivatives of the function
.
and
.
,
,
,
and
.
Here are the graphs of the partial derivatives :
Notice that here we have
, but this is not true in general.
Let's go back to the function
for all (
,
) except at (
,
), and
.
Since
and
.
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We have seen that
,
if (
) = (
) and
and
for all (
) other than (
,
). We get
and
.
In fact, we can also compute them directly from the definition :
.
and
.
It is not hard to see that
and
do not exist at (
,
).
Let's take a look at the function
for all (
,
) except at (
,
),and
.
Calculations, which we leave as an exercise, show that
and
.
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> diff((x^3*y-x*y^3)/(x^2+y^2),x):
simplify(%);
diff((x^3*y-x*y^3)/(x^2+y^2),y):
simplify(%);
![]()
![]()
Here are the graphs of
and
.
> x:='x':
y:='y':
diff(diff((x^3*y-x*y^3)/(x^2+y^2),y),x):
simplify(%);
diff(diff((x^3*y-x*y^3)/(x^2+y^2),x),y):
simplify(%);
![]()
![]()
Here are the graphs of
and
.
Note that
and
are not continuous at (
,
).
Clairaut's Theorem
Suppose that f is defined on a disk D that contains (
). If the functions
and
are continuous on D , then
.