calculator partial derivative - Aaron Graves, PhDude Replica

Function f(x, y)

Use explicit multiplication: type x*y, not xy. Supported: + - * / ^, sin, cos, tan, exp, log, sqrt, abs, pi, e.

x value

y value

Step size h (numerical precision)

Also evaluate f(x,y)

Quick examples:

What this calculator does

A partial derivative calculator helps you estimate how a multivariable function changes with respect to one variable while holding the others constant. For a function f(x, y), this tool computes:

∂f/∂x: the rate of change in the x-direction (y fixed)
∂f/∂y: the rate of change in the y-direction (x fixed)
Second-order partials: ∂²f/∂x², ∂²f/∂y², and mixed derivative ∂²f/(∂x∂y)

The calculator uses central finite differences, a standard numerical method that is usually accurate for smooth functions when the step size h is small.

How partial derivatives work

Intuition

Imagine a surface z = f(x, y). At a point (x, y), you can move a tiny bit in x or y and observe how z changes. Those directional changes become the partial derivatives.

Numerical formulas used here

\partialf/\partialx \approx [f(x+h, y) - f(x-h, y)] / (2h) \partialf/\partialy \approx [f(x, y+h) - f(x, y-h)] / (2h) \partial²f/\partialx² \approx [f(x+h, y) - 2f(x, y) + f(x-h, y)] / h² \partial²f/\partialy² \approx [f(x, y+h) - 2f(x, y) + f(x, y-h)] / h² \partial²f/(\partialx\partialy) \approx [f(x+h,y+h)-f(x+h,y-h)-f(x-h,y+h)+f(x-h,y-h)] / (4h²)

How to use this calculator

Enter a function in x and y, such as x^2*y + sin(x*y).
Enter a point (x, y) where you want the derivatives.
Choose a small step size h (default 0.0001 works well in many cases).
Click Calculate Partial Derivatives.

If you get unstable values, try a different h (for example 1e-5 or 1e-4). Very tiny h can amplify floating-point round-off error.

Supported math syntax

Operators: +, -, *, /, ^
Functions: sin, cos, tan, exp, log, sqrt, abs, asin, acos, atan
Constants: pi, e
Variables: x and y

Important: write multiplication explicitly. Use 2*x rather than 2x.

Example interpretation

Suppose f(x,y)=x^2*y + sin(x*y) at (1,2). If the calculator returns a positive ∂f/∂x, increasing x slightly (with y fixed) increases f. If ∂f/∂y is negative, increasing y slightly (with x fixed) decreases f near that point.

The mixed derivative ∂²f/(∂x∂y) tells you how the x-slope changes as y changes (or equivalently how the y-slope changes as x changes). In smooth functions, these are usually equal in either order.

Common mistakes and troubleshooting

1) Domain errors

Expressions like log(x) with x ≤ 0 or sqrt(x) with x < 0 produce invalid results. Move to a valid point or change the function.

2) Step size too large or too small

Too large: derivative estimate is coarse.
Too small: round-off noise can dominate.

3) Missing multiplication

xy is not the same as x*y. Always include *.

Why this matters

Partial derivatives are central in optimization, machine learning, economics, engineering, and physics. Gradient-based methods rely on these rates of change to find minima, maxima, and stable operating points. Even when symbolic differentiation is hard, numerical partial derivatives are practical and fast.