Partial Derivative Calculator

What is a partial derivative?

For a function of several variables, a partial derivative measures how the function changes when one variable moves and the others are held fixed. Notation: ∂f/∂x or fₓ. A partial differentiation calculator applies the same derivative rules you already know — power, product, quotient, chain — while treating non-selected letters as constants.

Example: if f(x, y) = x²y + y³, then ∂f/∂x = 2xy and ∂f/∂y = x² + 3y². Enter x^2*y + y^3 in this partial derivative calculator, set the variable to x or y, and compare both results.

How to use this partial differentiation calculator

1. Enter a multivariable expression (e.g. sin(x*y) or e^(x+y)).
2. Pick the differentiation variable in the toolbar (x, y, z, …).
3. Press Calculate to get ∂f/∂(variable) with labeled steps.
4. Change the variable and recalculate to see the other partial.

Any letter you did not select is held constant. Use * between multiplied factors (x*y) for clarity; implicit multiplication is accepted in many cases.

Partial derivative examples to practice

These examples work well as a free partial differentiation solver for homework checks:

Two-variable surfaces

• x^2*y + y^3 — polynomial surface; compare ∂/∂x and ∂/∂y
• sin(x*y) — product inside trig (chain rule)
• (x^2 + y)/(x - y) — quotient rule with two variables

Exponentials, logs, and three variables

• e^(x+y) — multivariable exponential
• ln(x^2 + y^2) — log of a sum of squares
• x*y*z — three variables; freeze two at a time

Partial vs ordinary derivatives

When to use an ordinary derivative

Ordinary derivatives (single-variable) assume one independent variable. If you only need d/dx of a function of x, use the main derivative calculator instead.

When partials are the right tool

Partial derivatives freeze every other variable. The algebra looks familiar — the difference is which symbols you treat as constants. This page is built for multivariable calculus: gradients piece together from partials, tangent planes use first partials, and many physics / economics models are written as f(x, y) or f(x, y, z).

Mixed partials and higher order

How to compute a mixed partial

To compute a mixed partial such as ∂²f/∂y∂x, first differentiate with respect to x, then paste the result and differentiate with respect to y (or raise the order after fixing one variable).

Clairaut’s theorem check

For smooth functions, mixed partials agree regardless of order — use this calculator to verify both paths on practice problems.

Related calculators