Partial Derivative Calculator: ∂f/∂x, ∂f/∂y, Mixed & Gradient

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Formula at a glance

Partial Derivative Calculator: Partial derivative — ∂f/∂x = lim (h→0) [f(x + h, y) − f(x, y)]/h. Example: ∂/∂x of x²y + sin(xy) → Partial derivative: ∂f/∂x = 2xy + y·cos(xy) — How the Partial Derivative Calculator works: Partial derivative — ∂f/∂x = lim (h→0) [f(x + h, y) − f(x, y)]/h

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📘 About the Partial Derivative Calculator

A partial derivative measures how a function of several variables changes when only one of them changes. ∂f/∂x treats y, z … as constants. Partial derivatives give gradients, tangent planes, rates in thermodynamics and economics (marginal cost and utility), and are the basis of multivariable optimisation.

Enter the function and the variable (or several letters for a higher or mixed derivative). The table lists all first partials and second partials, including the mixed ones, which are equal for smooth functions (Clairaut’s theorem). Give a point to get the value and the gradient vector there.

🧮 Formula

Partial derivative

∂f/∂x = lim (h→0) [f(x + h, y) − f(x, y)]/h

Gradient

∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)

Clairaut

∂²f/∂x∂y = ∂²f/∂y∂x

✏️ Worked examples

∂/∂x of x²y + sin(xy)

You enter
Function f(x, y, z …)	x^2y + sin(xy)
Differentiate with respect to	x

You get
Partial derivative	∂f/∂x = 2xy + y·cos(xy)
Function	f = x²y + sin(xy)

∂f/∂x = 2xy + y·cos(xy).

Mixed partial at a point

You enter
Function f(x, y, z …)	x^2y^3z
Differentiate with respect to	xy
Evaluate at (optional)	x=1, y=2, z=3

You get
Partial derivative	∂²f/∂y∂x = 6xy²z
Function	f = x²y³z
Value at the point	∂²f/∂y∂x = 72
Gradient ∇f at the point	(48, 36, 8)

∂²f/∂y∂x = 6xy²z; value ∂²f/∂y∂x = 72.

🧭 How to use this calculator

What you enter

Function f(x, y, z …)
Differentiate with respect toType the letters in order: xy = ∂/∂x then ∂/∂y.
Evaluate at (optional)

What you get

Partial derivative
Function

❓ Frequently asked questions

How do you take a partial derivative?

Treat every other variable as a number and differentiate as usual. For f = x²y, ∂f/∂x = 2xy and ∂f/∂y = x².

What is a mixed partial derivative?

Differentiating by one variable and then another, like ∂²f/∂x∂y. For smooth functions the order does not matter.

What is the gradient?

The vector of all first partial derivatives. It points in the direction of steepest increase, and its length is the rate of increase.

∂ Partial Derivative Calculator