The partial derivative of a function f with respect to the variable x is represented as f\x or (where is a rounded 'd' known as the 'partial derivative symbol').
If f is a function of x1, ..., xn and dx1, ..., dxn are thought of as infinitely small increments of x1, ..., xn respectively, then the corresponding infinitely small increment of f is
As an example, consider the volume V of a cone; it depends on the cone's height h and its radius r according to the formula
Equations involving an unknown function's partial derivatives are called partial differential equations and are ubiquitous throughout science.
Formal definition and properties
Like ordinary derivatives, the partial derivative is defined as a limit. Let U be an open subset of Rn and f : U -> R a function. We define the partial derivative of f at the point a=(a1,...,an)∈U with respect to the i-th variable xi as
The partial derivative ∂f/∂xi can be seen as another function defined on U and can again be partially differentiated. If all mixed partial derivatives exist and are continuous, we call f a C2 function; in this case, the partial derivatives can be exchanged:
See also: Directional derivative