Notation For Differentiation - Partial Derivatives

Partial Derivatives

When more specific types of differentiation are necessary, such as in multivariate calculus or tensor analysis, other notations are common.

For a function f(x), we can express the derivative using subscripts of the independent variable:

This is especially useful for taking partial derivatives of a function of several variables.

Partial derivatives will generally be distinguished from ordinary derivatives by replacing the differential operator d with a "∂" symbol. For example, we can indicate the partial derivative of f(x,y,z) with respect to x, but not to y or z in several ways:

where the final two notations are equivalent in flat Euclidean Space but are different in other manifolds.

Other generalizations of the derivative can be found in various subfields of mathematics, physics, and engineering.