partial derivative calculator - Aaron Graves, PhDude Replica

Instant Partial Derivative Calculator

Evaluate numerical partial derivatives for a two-variable function f(x, y) at a specific point using the central-difference method.

Function f(x, y)

Use x and y. Supported functions: sin, cos, tan, log, sqrt, exp, abs. Use ^ for powers (e.g., x^3).

x value

y value

Derivative to compute

Step size h (small positive number)

Quick examples:

What Is a Partial Derivative?

A partial derivative tells you how a multivariable function changes when you vary one input while holding the others fixed. If your function is f(x, y), then:

∂f/∂x measures change in the x-direction while y is constant.
∂f/∂y measures change in the y-direction while x is constant.

This concept appears everywhere: optimization, machine learning gradients, economics sensitivity analysis, thermodynamics, and engineering design.

How This Calculator Works

This tool computes derivatives numerically using the central-difference approximation, which is generally more accurate than one-sided differences for smooth functions.

Formulas Used

∂f/∂x ≈ [f(x+h, y) - f(x-h, y)] / (2h)
∂f/∂y ≈ [f(x, y+h) - f(x, y-h)] / (2h)

You control h, the step size. A very large h can reduce accuracy; a very tiny h can introduce floating-point noise. The default h = 1e-5 works well for many functions.

Input Format Guide

Use Standard Math Syntax

Multiplication must be explicit: write 2*x, not 2x.
Use ^ for powers: x^2, (x+y)^3.
Use parentheses for grouping.

Functions and Constants

Functions: sin, cos, tan, asin, acos, atan, sqrt, log, ln, exp, abs, pow.
Constants: pi and e.

Tip: If your function includes a domain restriction (for example sqrt(x-y) or log(x)), make sure your chosen point and nearby values stay valid. Numerical derivatives evaluate points slightly around your input.

Worked Examples

Example 1: `f(x,y)=x^2y + sin(xy)` at (1,2)

Set x=1, y=2, and choose “Both (gradient).” The calculator returns approximate values for ∂f/∂x and ∂f/∂y at that point.

Example 2: `f(x,y)=exp(x*y)+x^3-y^2`

Great for seeing how exponential growth and polynomial terms combine. Try different points and observe how sensitivity changes dramatically across the surface.

Common Mistakes to Avoid

Typing 2x instead of 2*x.
Using an invalid point (e.g., log(x) near non-positive values).
Choosing a step size h that is too large or negative.
Forgetting that this calculator is numerical, not symbolic.

Why Numerical Partial Derivatives Matter

Even when symbolic differentiation is available, numerical derivatives are valuable for quick checks, black-box models, simulation outputs, and data-driven functions where algebraic formulas are difficult to manipulate directly.

If you are learning multivariable calculus, this tool also helps you build intuition: move around a surface and see how each direction changes independently.