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. Seeing both answers side by side is often the fastest way to catch a “forgot to treat y as constant” mistake.

Geometrically, first partials feed gradients, tangent planes, and directional derivatives. In applied courses they show up as marginal rates when two or more inputs change independently — price and quantity, temperature and pressure, or x and y on a surface. This page stays focused on computing those partials with transparent steps.

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. Wrap numerators, denominators, and exponents in parentheses when they contain more than one token — (x^2 + y)/(x - y) is safer than ambiguous spacing.

A useful study loop: predict which letters freeze before you calculate, then check the step list. If you expected y to vanish from a power of x and it did not, you probably differentiated with respect to the wrong variable. Recalculate with the toolbar set correctly instead of rewriting the whole expression.

Partial derivative examples to practice

These examples work well as a free partial differentiation solver for homework checks. Run each twice — once in x and once in y (or z) — and write both answers before peeking at simplification details.

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
• x^3*y^2 — power rule on each factor while the other stays constant
• cos(x + y) — sum inside trig; both partials involve −sin

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
• e^(x*y) — product in the exponent (chain + product awareness)
• x^2*y*z + y*z — mixed polynomial in three letters

After the warm-ups, try an expression that mixes a quotient with a product, such as (x*y)/(x + y) or y*sin(x*z). The partial derivative calculator should show familiar single-variable rules with the “other letters are constant” twist made explicit in the working.

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. Switching tools keeps the interface and mental model aligned with the problem: no extra variable picker when nothing needs freezing.

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).

If a problem says “treat y as constant” or writes ∂ instead of d, stay here. If every letter other than x is a fixed parameter (a, b, k), you can still use this multivariable derivative calculator and differentiate with respect to x — parameters behave like the frozen variables in a partial.

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). Keep a short note of the path you took — “x then y” versus “y then x” — so you can compare both results on the same function.

Higher-order single-variable partials (∂²f/∂x², ∂³f/∂y³) are the same idea with the variable held fixed and the order raised. Use them for Hessian diagonal entries, concavity along one axis, or Taylor expansions in several variables.

Clairaut’s theorem check

For smooth functions, mixed partials agree regardless of order — use this calculator to verify both paths on practice problems. When the two paths disagree, recheck algebra and domain assumptions before blaming the theorem; most homework functions are smooth on the region you care about, so a mismatch usually means a differentiation slip.

Tips for clear multivariable input

Write every product with * when letters sit next to each other, and prefer ^ for powers. Set the differentiation variable explicitly every time you switch from ∂/∂x to ∂/∂y — leaving the previous setting is a common source of “wrong” answers that are actually the other partial. Start at order 1 unless the problem asks for a second partial or a mixed path.

When checking a gradient, compute each component separately and assemble ⟨∂f/∂x, ∂f/∂y⟩ (or with z) on paper. The free partial derivative calculator is fastest component-wise; you stay responsible for the vector packaging your course expects.

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