Partial Derivative Calculator: Partial Differentiation | ∂F

A versatile partial derivative calculator for partial differentiation of any order for real-valued multi-variable functions with respect to any of their independent variables, as well as (ordinary) derivatives of any order of one-variable functions.

It's easy to use this ordinary and partial derivative calculator:

  • Type in a real-valued function F with any number of arbitrary variables (e.g., F(x,y,z) = xyz), and also a variable in the boxes provided and press the Calculate Derivative button.
  • The derivative (such as F'(x)) or the partial derivative (such as ∂F/∂x) of the function with respect to the specified variable will be displayed in a newly added panel.
    • You can also calculate the ordinary or mixed partial derivatives of second and higher orders of the derived functions or of their primitives with respect to any variable by proceeding as above on the relevant panel.

You can close any panel by pressing × on that panel.

This multivariable derivative calculator for real-valued function, displays the calculated derivative (ordinary or partial) in a format that allows you to trace the steps of the differentiation, reflecting the applied rules of differentiation.

Note: The notation, for example, Fxy, represents the second order partial derivative of F with respect to x, and then with respect to y. It's also denoted by ∂2F/∂y∂x.

For a single variable function, Fx represents dF/dx or F'(x), and Fxx represents F"(x) or d2F/dx2.

F =
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Real-Valued Functions: Ordinary derivatives vs Partial Derivatives

The key difference lies in the number of independent variables: an ordinary derivative or just the derivative is calculated for a single-variable function, while a partial derivative is calculated for a multivariable function with respect to any of its independent variables. Note that when calculating partial derivatives, you treat all other variables as constants.

An example of an ordinary derivative is merely the derivative of the function f(x) = x^2, which is f'(x) = 2x. An example of a partial derivative is the partial derivative of the function g(x,y) = xy with respect to x, which is ∂g/∂x = y (treating y as a constant).

Since a partial derivative calculator can handle functions with any number of variables, it is inherently an ordinary derivative calculator or simply a derivative calculator for one-variable functions as well. This means it can calculate derivatives of single-variable functions like f(x) with respect to its only variable x, namely f'(x).