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Calculator Answers

Partial Derivative Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A partial derivative calculator is a mathematical utility that computes the derivative of a multi-variable function with respect to one variable while treating all other variables as constant values. It parses algebraic expressions symbolically using computer algebra systems and applies standard calculus rules. The calculator can also evaluate the resulting partial derivative at a specific coordinate point. It is widely used in physics, engineering, optimization problems, and machine learning algorithms such as gradient descent to analyze rate of change across multiple dimensions.

Compute symbolic partial derivatives of multi-variable functions. Select the target variable and optionally evaluate the derivative at specific coordinate points.

See also: Partial Fraction Decomposition Calculator, Derivative Calculator, and Double Integral Calculator.

Quick Answer

Compute symbolic partial derivatives of multi-variable functions. Select the independent variable and optionally evaluate the derivative at specific coordinate points.

Use standard algebraic syntax (e.g. x^2 * y, sin(x * y)).

Select the variable to treat as independent. All other variables are treated as constants.

Evaluate At Point (Optional)

Leave field empty to calculate the symbolic derivative only.

Partial Derivative

∂f/∂x =

2 · y · x + 3 · y ^ 2

Evaluated at (x = 2, y = 3) = 39

Function fx²·y + 3·x·y²
Respect tox
∂f/∂x2 · y · x + 3 · y ^ 2
Point Evaluated(x = 2, y = 3)
f′(x) at point39

Example: ∂/∂x of (x²y + 3xy²) = 2xy + 3y²

Other variables are treated as constant. Symbolic computation is performed using Math.js.

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Examples

∂/∂x of (x²y + 3xy²)

∂f/∂x = 2xy + 3y²

∂/∂y of (x²y + 3xy²)

∂f/∂y = x² + 6xy

∂/∂x of sin(x * y) at (x=1, y=π)

∂f/∂x = y * cos(x * y) · Evaluated = π * cos(π) = -π

How it works

The calculator takes a multi-variable function, parses it into an expression tree, and applies symbolic differentiation rules with respect to the chosen variable.

Formal Limit Definition

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

Core Principles

  • Treat the variable of interest as the only independent variable.
  • Treat all other symbols as fixed numeric constants.
  • Apply standard derivative rules (power rule, product rule, chain rule).

How Partial Differentiation Works

When dealing with functions of more than one variable (such as f(x,y) = x2y + 3xy2), we often want to know how the function changes with respect to one variable at a time:

  • Differentiating with respect to x: We hold y constant. The term x2y becomes 2xy, and the term 3xy2 becomes 3y2. Adding these together gives the partial derivative: ∂f/∂x = 2xy + 3y2.
  • Differentiating with respect to y: We hold x constant. The term x2y becomes x2, and the term 3xy2 becomes 6xy. Adding these together yields: ∂f/∂y = x2 + 6xy.

Applications of Partial Derivatives

Partial derivatives are central to multivariable calculus, physics, and machine learning:

  • Gradient Vector: The collection of all first-order partial derivatives (∇f = ⟨∂f/∂x, ∂f/∂y⟩) points in the direction of steepest ascent on a surface. This is the foundation of Gradient Descent in AI.
  • Physics and Thermodynamics: Used to model fluid flow, electromagnetism, and thermodynamic state changes, where variables like pressure, volume, and temperature depend on one another.

Related Mathematical Tools

If you are working with single-variable functions, you can use our standard Derivative Calculator or look into integration using the Integral Calculator.

Related Calculators

More tools from Education

Partial Fraction Decomposition CalculatorDecompose (ax + b) / ((x − r₁)(x − r₂)) into A / (x − r₁) + B / (x − r₂) for distinct linear factors. Returns A, B, and the decomposed form with step notes.Derivative CalculatorDifferentiate polynomials term by term with the power rule — clear breakdown of each term's derivative.Double Integral CalculatorCalculate the definite double integral of a function of two variables over rectangular regions using numerical integration.Rate of Change CalculatorCompute either the average rate of change of a function over an interval or the instantaneous rate of change at a specific point.

Frequently asked questions

A partial derivative of a multi-variable function is its derivative with respect to one variable, while all other variables are treated as constants. It measures the rate at which the function changes along the direction of that variable's coordinate axis.

To calculate a partial derivative, choose the target variable (such as x) and differentiate the function normally using ordinary derivative rules (power rule, chain rule, product rule, etc.), treating all other variables (like y, z) as if they were fixed numbers.

Partial derivatives are denoted using the curly 'del' symbol (∂). The partial derivative of f with respect to x is written as ∂f/∂x, ∂/∂x f(x, y), or fx.

Yes, you can optionally enter specific values for your variables (e.g. x = 2, y = 3) to compute the numeric slope of the function at that exact coordinate point.

Geometrically, a multi-variable function represents a surface. The partial derivative with respect to x at a point is the slope of the line tangent to the surface parallel to the x-axis.