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Partial Derivative Calculator
Last updated: May 31, 2026
Written by Blake Boege
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.
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.
∂f/∂x =
2 · y · x + 3 · y ^ 2
Evaluated at (x = 2, y = 3) = 39
Example: ∂/∂x of (x²y + 3xy²) = 2xy + 3y²
Other variables are treated as constant. Symbolic computation is performed using Math.js.
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 $f(x,y) = x^2 y + 3xy^2$), 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 $x^2 y$ becomes $2xy$, and the term $3xy^2$ becomes $3y^2$. Adding these together gives the partial derivative: $\partial f/\partial x = 2xy + 3y^2$.
- Differentiating with respect to y: We hold x constant. The term $x^2 y$ becomes $x^2$, and the term $3xy^2$ becomes $6xy$. Adding these together yields: $\partial f/\partial y = x^2 + 6xy$.
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 ($\nabla f = \langle \partial f/\partial x, \partial f/\partial y \rangle$) 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.
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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.
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