Partial Derivative Calculator — Free Online

The Partial Derivative Calculator computes partial derivatives of functions with multiple variables. It differentiates with respect to your chosen variable while treating all other variables as constants.

Enter a multivariable function (e.g. x^2*y + sin(y)) and specify which variable to differentiate with respect to. The calculator uses symbolic differentiation to give exact results.

Function f(x, y)

Differentiate with respect to

Your calculations will appear here

How to Use the Partial Derivative Calculator

Enter a function of multiple variables (e.g. x^2*y + sin(y)).
Select the variable to differentiate with respect to.
All other variables are treated as constants during differentiation.

Formula and Theory

A partial derivative measures how a multivariable function changes when only one variable changes, while all other variables are held constant. It is denoted with the symbol ∂ (partial).

Partial derivatives are foundational in multivariable calculus, used in gradient vectors, directional derivatives, tangent planes, and optimisation of functions with multiple variables.

Worked Examples

Example 1: Simple partial derivative

Problem: Find df/dx of f(x,y) = x^2*y + 3y^2

Solution: Treat y as constant: d/dx(x^2*y) = 2xy, d/dx(3y^2) = 0

Answer: df/dx = 2xy

Example 2: Partial with respect to y

Problem: Find df/dy of f(x,y) = x^2*y + 3y^2

Solution: Treat x as constant: d/dy(x^2*y) = x^2, d/dy(3y^2) = 6y

Answer: df/dy = x^2 + 6y

Frequently Asked Questions

What is the gradient vector?

The gradient is a vector of all partial derivatives: grad(f) = (df/dx, df/dy, ...). It points in the direction of steepest increase of the function.

What does the notation df/dx vs d/dx mean for partial derivatives?

The curly d symbol (∂f/∂x) denotes a partial derivative, indicating that other variables are held constant. The straight d (df/dx) is used for ordinary (single-variable) derivatives.