Partial Derivative Calculator

A partial derivative measures how a multivariable function changes with respect to one variable while all other variables are held constant. For example, ∂f/∂x of f(x,y) treats y as a constant and differentiates only with respect to x. The notation ∂ (the partial symbol) distinguishes it from ordinary derivatives.

Type your function using standard math notation: use * for multiplication (e.g. x*y), ^ for exponentiation (e.g. x^2), and named functions like sin(), cos(), exp(), log(), sqrt(). For example, enter x^2*y + sin(x*y) to compute the partial derivative of x²y + sin(xy).

A regular (ordinary) derivative applies to functions of a single variable, while a partial derivative applies to functions of multiple variables. When computing a partial derivative with respect to x, all other variables such as y and z are treated as constants, unlike the chain rule treatment they would receive in total differentiation.

A higher-order partial derivative is obtained by differentiating a function more than once. For example, the second-order partial derivative ∂²f/∂x² differentiates f twice with respect to x. Mixed partial derivatives like ∂²f/∂x∂y differentiate first with respect to one variable, then another.

Mixed partial derivatives involve differentiating with respect to two different variables, such as ∂²f/∂x∂y. By Clairaut's theorem, if both mixed partial derivatives are continuous, then ∂²f/∂x∂y = ∂²f/∂y∂x — the order of differentiation does not matter.

Partial derivatives are used extensively in physics (gradient, heat equation, wave equation), economics (marginal utility, cost optimization), machine learning (gradient descent for training neural networks), engineering (stress analysis, fluid dynamics), and thermodynamics (relating state variables like pressure, volume, and temperature).

This calculator supports polynomial expressions (x^2, y^3), trigonometric functions (sin, cos, tan), inverse trig (asin, acos, atan), exponential functions (exp, e^x), natural logarithm (log or ln), and square roots (sqrt). Combine them freely with +, -, *, /, and ^ operators.

Geometrically, the partial derivative ∂f/∂x at a point (a, b) gives the slope of the tangent line to the surface z = f(x,y) in the direction parallel to the x-axis at that point. It tells you the rate of change of the function as you move only in the x-direction, keeping y fixed.