Calculator Input
Enter a function in x and y. Use explicit multiplication such as x*y or 2*x^2*y. Supported functions include sin, cos, tan, exp, log, ln, log10, sqrt, and abs.
Example Data Table
This sample illustrates a typical multivariable function evaluated numerically with a central-difference step size.
| Function | x | y | h | ∂f/∂x | ∂f/∂y | ∂²f/∂x∂y |
|---|---|---|---|---|---|---|
| x^2*y + sin(x*y) | 1 | 2 | 0.001 | 3.1677068818 | 0.5838532328 | -0.2347391344 |
| exp(x*y) + x^3 - y^2 | 0.5 | 1.2 | 0.001 | 2.786531 | 0.511055 | 2.915744 |
| log(x + y + 3) + x*y^2 | 1.5 | 0.8 | 0.0005 | 1.761905 | 2.552381 | 1.980952 |
Formula Used
First partial with respect to x
∂f/∂x ≈ [f(x+h, y) - f(x-h, y)] / 2h
First partial with respect to y
∂f/∂y ≈ [f(x, y+h) - f(x, y-h)] / 2h
Second partials
∂²f/∂x² ≈ [f(x+h, y) - 2f(x, y) + f(x-h, y)] / h²
∂²f/∂y² ≈ [f(x, y+h) - 2f(x, y) + f(x, y-h)] / h²
Mixed partial derivative
∂²f/∂x∂y ≈ [f(x+h, y+h) - f(x+h, y-h) - f(x-h, y+h) + f(x-h, y-h)] / 4h²
Clairaut symmetry check
For smooth functions, ∂²f/∂x∂y and ∂²f/∂y∂x should be approximately equal near the same point.
Extra metrics
Gradient magnitude = √[(∂f/∂x)² + (∂f/∂y)²], Laplacian = ∂²f/∂x² + ∂²f/∂y², Hessian determinant = (∂²f/∂x²)(∂²f/∂y²) - (∂²f/∂x∂y)(∂²f/∂y∂x).
How to Use This Calculator
- Enter a multivariable function using x and y.
- Type the evaluation point coordinates in the x and y fields.
- Choose a step size h for numerical differentiation.
- Press Calculate Derivatives to generate the result set.
- Review the mixed partials, first partials, and second partials.
- Use the symmetry gap to compare fxy and fyx.
- Download the result table as CSV or PDF when needed.
- Use explicit multiplication symbols to avoid parser errors.
Frequently Asked Questions
1. What does the calculator actually compute?
It numerically estimates first partial derivatives, second partial derivatives, mixed partial derivatives, gradient magnitude, Laplacian, Hessian determinant, and a symmetry check at a chosen point.
2. Why are fxy and fyx usually the same?
For smooth functions with continuous second derivatives, Clairaut's theorem says the mixed partials should match. Small differences usually come from numerical rounding or an unsuitable step size.
3. Which functions are supported in the input?
The parser supports x, y, constants pi and e, arithmetic operators, powers, parentheses, and common functions such as sin, cos, tan, exp, log, ln, log10, sqrt, and abs.
4. Do I need to type multiplication signs explicitly?
Yes. Write expressions like 2*x*y or x^2*y. Implicit multiplication, such as 2xy or x(y+1), is not interpreted by this parser.
5. How should I choose the step size?
A moderate step often works best. Very large steps reduce accuracy, while extremely tiny steps amplify rounding errors. Values near 0.001 or 0.0001 are good starting points.
6. Is this symbolic differentiation?
No. This page uses numerical central-difference formulas. It estimates derivatives near the chosen point instead of generating an exact algebraic derivative expression.
7. What does the Hessian determinant tell me?
It summarizes local curvature information from second derivatives. Together with other second-order values, it helps classify nearby behavior, including saddle-like or bowl-like tendencies.
8. Why might the calculator show an error?
Errors appear when the expression has invalid syntax, unsupported symbols, mismatched parentheses, impossible operations like log of a negative value, or a step size that is too small.