Just as we had higher order derivatives with functions of
one variable we will also have higher order derivatives of functions of more
than one variable. However, this time we
will have more options since we do have more than one variable..
Consider the case of a function of two variables, since both of the first order partial
derivatives are also functions of x
and y we could in turn differentiate
each with respect to x or y.
This means that for the case of a function of two variables there will
be a total of four possible second order derivatives. Here they are and the notations that we’ll
use to denote them.
The second and third second order partial derivatives are
often called mixed partial derivatives since we are taking derivatives with
respect to more than one variable. Note
as well that the order that we take the derivatives in is given by the notation
for each these. If we are using the
subscripting notation, e.g. ,
then we will differentiate from left to right.
In other words, in this case, we will differentiate first with respect
to x and then with respect to y.
With the fractional notation, e.g.
it is the opposite. In these cases we
differentiate moving along the denominator from right to left. So, again, in this case we differentiate with
respect to x first and then y.
Let’s take a quick look at an example.
Example 1 Find
all the second order derivatives for .
We’ll first need the first order derivatives so here they
Now, let’s get the second order derivatives.
Notice that we dropped the from the derivatives. This is fairly standard and we will be doing
it most of the time from this point on.
We will also be dropping it for the first order derivatives in most
Now let’s also notice that, in this case, . This is not by coincidence. If the function is “nice enough” this will
always be the case. So, what’s “nice
enough”? The following theorem tells us.
Now, do not get too excited about the disk business and the fact
that we gave the theorem for a specific point. In pretty much every example in this class if
the two mixed second order partial derivatives are continuous then they will be
Example 2 Verify
Clairaut’s Theorem for .
We’ll first need the two first order derivatives.
Now, compute the two fixed second order partial
Sure enough they are the same.
So far we have only looked at second order derivatives. There are, of course, higher order
derivatives as well. Here are a couple
of the third order partial derivatives of function of two variables.
Notice as well that for both of these we differentiate once
with respect to y and twice with
respect to x. There is also another third order partial
derivative in which we can do this, . There is an extension to Clairaut’s Theorem
that says if all three of these are continuous then they should all be equal,
To this point we’ve only looked at functions of two
variables, but everything that we’ve done to this point will work regardless of
the number of variables that we’ve got in the function and there are natural
extensions to Clairaut’s theorem to all of these cases as well. For instance,
provided both of the
derivatives are continuous.
In general, we can extend Clairaut’s theorem to any function
and mixed partial derivatives. The only
requirement is that in each derivative we differentiate with respect to each
variable the same number of times. In
other words, provided we meet the continuity condition, the following will be
because in each case we differentiate with respect to t once, s three times and r three
Let’s do a couple of examples with higher (well higher order
than two anyway) order derivatives and functions of more than two variables.