Second Partial Derivative

Second Partial Derivative:

Functions of More Than Two Variables:

The definitions of the partial derivatives of functions of more than two independent variables are like the definitions for functions of two variables. They are ordinary derivatives with respect to one variable, taken while all the other independent variables are regarded as constants.
Example:
is a function of three variables,
then we can define the partial derivatives in much the same way
as we defined the partial derivatives for three variables. So, we have:

and
Second Partial Derivative:
When we differentiate a function f (x, y) twice, we produce its second order derivatives. These derivatives are usually denoted as


Differentiate first with respect to y, then
with respect to x. And


Example:
find: If



Solution:







Example:
find: If

Solution:




Exercise:
find 1) If

find 2) If