Calculator Inputs
Example Data Table
| Expression | x | y | z | h | Purpose |
|---|---|---|---|---|---|
| x^2*y + sin(x*y) + z^2 | 1.2 | 0.8 | 2.0 | 0.001 | General mixed partial and curvature test. |
| exp(x*y) + log(z + 3) | 0.5 | 1.4 | 1.0 | 0.0005 | Growth, coupling, and domain-sensitive behavior. |
| sqrt(x^2 + y^2 + z^2) + cos(x*z) | 2.0 | 1.0 | 0.5 | 0.001 | Radial change plus oscillatory interaction. |
Formula Used
First partial derivative
∂f/∂x ≈ [f(x+h,y,z) - f(x-h,y,z)] / 2h
Second partial derivative
∂²f/∂x² ≈ [f(x+h,y,z) - 2f(x,y,z) + f(x-h,y,z)] / h²
Mixed partial derivative
∂²f/∂x∂y ≈ [f(x+h,y+h,z)-f(x+h,y-h,z)-f(x-h,y+h,z)+f(x-h,y-h,z)] / 4h²
Derived measures
Gradient magnitude = √[(fx)² + (fy)² + (fz)²]. Laplacian = fxx + fyy + fzz.
Central differences are accurate for smooth functions when h is small but not excessively tiny. Very small h can amplify rounding error.
How to Use This Calculator
- Enter a valid multivariable expression using x, y, and optionally z.
- Set the evaluation point where you want the derivatives.
- Choose a step size h. Start with 0.001 for most smooth functions.
- Set graph span and grid points to control the Plotly surface resolution.
- Press the solve button to display the results above the form.
- Use the export buttons to save the solved metrics as CSV or PDF.
FAQs
1. Does this calculator perform symbolic differentiation?
No. It estimates partial derivatives numerically using central finite differences. This makes it flexible for many expressions, but results depend on the chosen step size and function smoothness.
2. What variables can I use?
You can use x, y, and z. If your function only needs two variables, keep z at any convenient constant value, such as 0.
3. Which functions are supported in expressions?
Supported functions include sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, exp, log, ln, sqrt, abs, pow, pi, and e.
4. Why is explicit multiplication required?
Write 2*x instead of 2x. Explicit multiplication avoids parsing ambiguity and keeps evaluation safer and more consistent for user-entered formulas.
5. How should I choose the step size h?
Use a small positive value, often 0.001 or 0.0005. Too large reduces accuracy, while too small may increase floating-point noise.
6. What does the Laplacian tell me?
The Laplacian adds the pure second partial derivatives. It summarizes local curvature and appears often in physics, diffusion, optimization, and field analysis.
7. What does the surface plot represent?
It shows how the function changes over x and y near your chosen point while keeping z fixed. This helps you inspect slope, curvature, and nearby shape.
8. Can I export my results?
Yes. The page includes CSV export for tabular values and PDF export for the visible solved report, making sharing and record keeping easier.