2 Second partial derivatives
Performing two successive partial differentiations of with respect to (holding constant) is denoted by (or ) and is defined by
For functions of two or more variables as well as other second-order partial derivatives can be obtained. Most obvious is the second derivative of with respect to is denoted by (or ) which is defined as:
Example 5
Find and for .
Solution
.
We can use the alternative notation when evaluating derivatives.
Example 6
Find and for .
Solution
2.1 Mixed second derivatives
It is possible to carry out a partial differentiation of with respect to followed by a partial differentiation with respect to (or vice-versa). The results are examples of mixed derivatives . We must be careful with the notation here.
We use to mean ‘differentiate first with respect to and then with respect to ’ and we use to mean ‘differentiate first with respect to and then with respect to ’:
(This explains why the order is opposite of what we expect - the derivative ‘operates on the left’.)
Example 7
For find
Solution
The remaining possibility is to differentiate first with respect to and then with respect to i.e. .
For the function in Example 7 and Notice that for this function
This equality of mixed derivatives is true for all functions which you are likely to meet in your studies.
To evaluate a mixed derivative we can use the alternative notation. To evaluate we write to indicate that the first differentiation is with respect to . Similarly, is denoted by .
Example 8
Find for the function
Solution
and
Task!
Find for