A partial derivative is a derivative of a function of several variables with respect to one variable, holding the others fixed.
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(1)
|
The above partial derivative is sometimes denoted 
for brevity.
Repeated partial derivatives are denoted by subscripts recording the order of differentiation from left to right. The operators in Leibniz notation act from right to left (Lebl 2026, p. 60), so
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
Such partial derivatives involving more than one variable are called mixed partial derivatives.
For a twice continuously differentiable function 
in a neighborhood of
,
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(5)
|
More generally, if all partial derivatives of order at most 
are continuous, their
order can be interchanged. For 
this gives
 |
(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, Trott 2006). 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.
