Let
be a function of
and
;
for example

The *partial derivative of
with respect to
*
is the function
obtained by differentiating
with respect to
,
treating
as a constant; in this case

The *partial derivative of
with respect to
*
is the function
obtained by differentiating
with respect to
,
treating
as a constant; in this case

These partial derivatives are formally defined as limits:

**Example 1.** For

we have

and

**Example 2.** For the function

we have

The number

gives the rate of change with respect to
of the function

the number

gives the rate of change with respect to
of the function

Through the surface we pass a plane parallel to the -plane. The plane intersects the surface in a curve, the -section of the surface.

The
-section
of the surface is the graph of the function

Differentiating with respect to
,
we have

and in particular

*The number
is thus the slope of the
-section
of the surface
at the point
*

The other partial derivative can be given a similar interpretation. The same surface is sliced by a plane parallel to the -plane. The plane intersects the surface in a curve, -section of the surface.

The
-section
of the surface is the graph of the function

Differentiating, this time with respect to
,
we have

and thus

*The number
is the slope of the
-section
of the surface
at the point
*

In the case of a function of three variables, there are three partial
derivatives: the partial with respect to
,
the partial with respect to
,
and also the partial with respect to
.
These partials

are defined as follows:

Each partial can be found by differentiating with respect to the subscript
variable, treating the other two variables as constants.

**Example 3.** For the function

the partial derivatives are

In particular

**Example 4.** For

we have

**Example 5.** For function of the form

we can write

*The number
gives the rate of change with respect to
of
at
;
gives the rate of change with respect to
of
at
and
gives the rate of change with respect to
of
at
*

**Example 6.** The function

has partial derivatives

The numbers
gives the rate of change with respect to
of the function

gives the rate of change with respect to
of the function

gives the rate of change with respect to
of the function

There is obviously no need to restrict ourselves to the variables Where convenient we can use other letters.

**Example 7.** The volume of the frustum of a cone is given by
the function

At time
,
Find the rate of change of the volume with respect to each of its dimensions
at time
if the other dimensions are held constant.

Solution. The partial derivatives of
are as follows:

At time
,

the rate of change of with respect to is

the rate of change of with respect to is

the rate of change of with respect to is

The subscript notation is not the only one used for partial differentiation. A
variant of Leibniz's double-d notation is also commonly used. In this notation
the partials
are denoted by

Thus, for

we have

or more simply,

We can also write

The ``double-decker'' notation is not restricted to the letters
For we can write

For the function

we have

Let
The function is said to have a **limit**
at
if for each
there exists
such that, if
then
In this case we write

**Example 1.** We will show that the function

**does not have a limit at **

Along the obvious paths to , the coordinate axes, the limiting value is :

along the -axis, and thus tends to ;

along the -axis, and thus tends to .

However, along the line
the limiting value is

We have shown that not all paths to yield the same limiting value. It follows that does not have a limit at

As in the one-variable case, the limit (if it exists) is unique. Moreover, if

then

and

To say that
is continuous at
is to say that

or, equivalently, that

For two variables we can write

and for three variables

To say that is continuous on is to say that is continuous at all points of .

Polynomials in several variables, for example,

are everywhere continuous. In the two-variable case, that means continuity at
each point of the
-plane;
in the three-variable case, continuity at each point of three-space.

Rational functions (quotients of polynomials) are continuous everywhere except
where the denominator is zero. Thus

is continuous at each point of the
-plane
other than the origin

is continuous except on the line

is continuous except on the parabola

is continuous at each point of three-space other than the origin

is continuous except on the plane
.

More elaborate continuous functions can be constructed by forming composites:
take, for example,

The first function is continuous except along the vertical plane
.
The other two functions are continuous at each point of space. The continuity
of such composites follows from a simple theorem that we state and prove
below. In the theorem,
is a function of several variables, but
is a function of a single variable.

**Theorem.** *If
is continuous at the point
and
is continuous at the number
then the composition
is continuous at the point
.*

Proof. We begin with
.
We must show that there exists
0 such that

From the continuity of
at
we know that there exists
such that

From the continuity of
at
,
we know that there exists
such that

This last
obviously works; namely,

A continuous function of several variables is continuous in each of its
variables separately. In the two-variable case, this means that, if

then

**The converse is false.**

**Example 2.** We set

Since

we have

Thus, at the point
,
is continuous in
and continuous in
.
However, as a function of two variables,
is not continuous at
One way to see this is to note that we can approach
as closely as we wish by points of the form
with
.
At such points
takes on the value
:

Hence,
cannot tend to
as required.

For functions of a single variable the existence of the derivative guarantees
continuity. For functions of several variables the existence of partial
derivatives fails to guarantee continuity. To show this, we can use the same
function

Since both
and
are constantly zero, both partials exist (and are zero) at
,
and yet, the function is discontinuous at

It is not hard to understand how a function can have partial derivatives and
yet fail to be continuous. The existence of
at
depends on the behavior of
only at points of the form
.
Similarly, the existence of
at
depends on the behavior of
only at points of the form
.
On the other hand, continuity at
depends on the behavior of
at points of the more general form
.
More briefly, we can put it this way: *the existence of a partial
derivative depends on the behavior of the function along a line segment (two
directions), whereas continuity depends on the behavior of the function in all
directions.*

Suppose that
is a function of
and
with first partials

These are again functions of
and
and may themselves possess partial derivatives:

These last functions are called the *second-order partials.* Note that
there are two ``mixed'' partials

The first of these is obtained by differentiating first with respect to
and then with respect to
.
The second is obtained by differentiating first with respect to
and then with respect to
.

**Example 3.** The function
has first partials

The second-order partials are

**Example 4.** Setting
we have

The second-order partials are

Notice that in both examples we had

Since in neither case was
symmetric in
and
,
this equality of the mixed partials was not due to symmetry. Actually it was
due to continuity.

**Theorem.** If
and its partials

are continuous then the mixed partials

Proof. Fix
Assume that

Then there exists
such that

for
From the fundamental theorem of calculus we have

Similarly,

Since the two iterated integrals are equal

we have

a contradiction.

In the case of a function of three variables we look for three first partials

and nine second partials

Here again, there is equality of the mixed partials

provided that
and its first and second partials are continuous.

**Example 5.** For

we have