Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest are held fixed during the differentiation.
 |
(1)
|
The above partial derivative is sometimes denoted 
for brevity.
Partial derivatives can also be taken with respect to multiple variables, as denoted for examples
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
Such partial derivatives involving more than one variable are called mixed partial derivatives.
For a "nice" two-dimensional function 
(i.e., one for which 
, 
, 
, 
, 
exist and are continuous in a neighborhood 
), then
 |
(5)
|
More generally, for "nice" functions, mixed partial derivatives must be equal regardless of the order in which the differentiation is performed, so it also is true that
 |
(6)
|
If the continuity requirement for mixed partials is dropped, it is possible to construct functions for which mixed partials are not equal. An example is the function
 |
(7)
|
which has 
and 
(Wagon 1991). This function is depicted above and by Fischer (1986).
Abramowitz and Stegun (1972) give finite difference versions for partial derivatives.
A differential equation expressing one or more quantities in terms of partial derivatives is called a partial differential equation. Partial differential equations are extremely important in physics and engineering, and are in general difficult to solve.
