Let Then
Section 2.3 Higher Order Derivatives
You have already observed, in your first Calculus course, that if is a function of then its derivative, is also a function of and can be differentiated to give the second order derivative which can in turn be differentiated yet again to give the third order derivative, and so on.
We can do the same for functions of more than one variable. If is a function of and then both of its partial derivatives, and are also functions of and They can both be differentiated with respect to and they can both be differentiated with respect to So there are four possible second order derivatives. Here they are, together with various alternate notations.
Example 2.3.1.
Example 2.3.2.
Let Then
More generally, for any integers
Example 2.3.3.
If then
and
In all of these examples, it didn’t matter what order we took the derivatives in. The following theorem shows that this was no accident.
1
The history of this important theorem is pretty convoluted. See “A note on the history of mixed partial derivatives” by Thomas James Higgins which was published in Scripta Mathematica 7 (1940), 59-62. The Theorem is named for Alexis Clairaut (1713--1765), a French mathematician, astronomer, and geophysicist, and Hermann Schwarz (1843--1921), a German mathematician.
Theorem 2.3.4. Clairaut’s Theorem or Schwarz’s Theorem.
Subsection 2.3.1 Optional — The Proof of Theorem 2.3.4
Subsubsection 2.3.1.1 Outline
Here is an outline of the proof of Theorem 2.3.4. The (numbered) details are in the subsection below. Fix real numbers and and define
We define in this way because both partial derivatives and are limits of as Precisely, we show in item (1) in the details below that
Note that the two right hand sides here are identical except for the order in which the limits are taken.
Subsubsection 2.3.1.2 The Details
- By definition,
-
The mean value theorem (Theorem 2.13.4 in the CLP-1 text) says that, for any differentiable function
- the slope of the line joining the points
and on the graph of
is the same as- the slope of the tangent to the graph at some point between
and
That is, there is some such thatApplying this with replaced by and replaced by givesHence, for some - Define
By the mean value theorem, - Define
By the mean value theorem, - Define
By the mean value theorem
Subsection 2.3.2 Optional — An Example of
In Theorem 2.3.4, we showed that if the partial derivatives and exist and are continuous at Here is an example which shows that if the partial derivatives and are not continuous at then it is possible that
Exercises 2.3.3 Exercises
Exercise Group.
Exercises — Stage 1
Exercise Group.
Exercises — Stage 2
3.
4.
Find all second partial derivatives of