Derivative Calculator | Ordinary & Partial Derivatives

About the Ordinary & Partial Derivative Calculator

Use our free online derivative calculator to compute symbolic ordinary or partial derivatives of real-valued functions with any number of independent variables with respect to any of those variables. Our Calculator Math™ derivative calculator supports mixed 2nd- and higher-order partial derivatives, such as F_xyz, as well as higher-order ordinary derivatives like the 2nd-order derivative F''(x).

Derivative Notations

The derivative 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.

Guide to Using Our Partial Derivative Calculator

Using this 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.

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:

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).