Partial Derivative Calculator: Higher-Order & Mixed Partials

Use our free, fast online partial derivative calculator to compute symbolic partial derivatives—such as ∂F/∂x—of multivariable functions with respect to any of their variables. Our derivative calculator supports higher-order mixed partial derivatives, such as F_xyz, as well as ordinary derivatives such as F'(x), F"(x), and more.

About the Derivative Calculator (Ordinary & Partial)

Our versatile partial derivative calculator allows you to calculate partial derivatives, such as F_x (also denoted as ∂f/∂x) for functions with any number of variables.

Since a partial differentiation 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, such as f'(x), f''(x), f⁽³⁾(x), etc.

Guide to Using Our Derivative Calculator

Using this ordinary and partial derivative calculator is straightforward:

Type in a function F with any number of arbitrary variables (e.g., xyz).
Type in a variable you want to differentiate the function with respect to.
Press the Calculate Derivative button.

The multivariable derivative calculator displays the calculated derivative (partial or ordinary) in a newly added panel. The derivative is presented in a format that allows you to trace the steps of differentiation, reflecting the applied rules of differentiation such as the product rule, quotient rule or chain rule.

Higher-Order & Mixed Partials: You can also compute second-order (and higher‑order) ordinary or mixed partial derivatives of the resulting functions by repeating steps 2 and 3 within the relevant result panel.

You can close any panel by pressing ×.

F =

with respect to:

Note: The derivative calculator is case-sensitive. Ensure the case of the variable you enter matches the case used in the function. This is especially important on mobile devices, where keyboards often auto-capitalize the first letter of an input.

Derivative Notations

The differentiation calculator uses standard notations for both ordinary and partial derivatives.

For example, F_xy, represents the mixed second-order partial derivative of F with respect to x, and then with respect to y. It's also denoted as ∂²F/∂y∂x.

For a single variable function, F_x represents dF/dx or F'(x), and F_xx represents the second-order derivative, F"(x) or d²F/dx², and so on.

Ordinary Derivatives vs Partial Derivatives

The key difference lies in the number of independent variables.

If a function has only one independent variable, its derivatives are called ordinary derivatives (with respect to that variable).
If a function has multiple independent variables, its derivatives are called partial derivatives, taken with respect to a specified variable. Note that when calculating partial derivatives, the partial differentiation calculator treats 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).