Estimate gradients, Hessians, and tangent planes fast with precision. Test directional change at chosen points. Understand local behavior with clear outputs and practical guidance.
Use x, y, and z. Supported functions include sin(), cos(), tan(), exp(), log(), sqrt(), and abs().
| Function | Point (x,y,z) | Direction | Step h | What to Inspect |
|---|---|---|---|---|
| x^2*y + sin(x*y) + z^2 | (1, 2, 1) | (1, 1, 0) | 0.0001 | Gradient, Hessian, directional rate |
| exp(x*y) + y*z + x^3 | (0.5, 1.2, -1) | (2, -1, 1) | 0.0001 | Mixed partial behavior |
| sqrt(x^2 + y^2 + z^2 + 1) | (2, -1, 3) | (1, 0, 1) | 0.0005 | Smooth radial change |
For a multivariable function f(x,y,z), the calculator estimates first partial derivatives with the central difference rule:
∂f/∂x ≈ [f(x+h,y,z) − f(x−h,y,z)] / (2h)
∂f/∂y ≈ [f(x,y+h,z) − f(x,y−h,z)] / (2h)
∂f/∂z ≈ [f(x,y,z+h) − f(x,y,z−h)] / (2h)
Second derivatives use central second differences, while mixed partials use four-point stencils. The gradient is:
∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)
The directional derivative along a unit vector u is:
Duf = ∇f · u
For a two-variable surface view, the tangent plane near (x₀,y₀) is approximated by f(x₀,y₀) + fₓ(x−x₀) + fᵧ(y−y₀).
x, y, and optional z.0.0001.It estimates first partial derivatives, second partial derivatives, mixed partials, gradient magnitude, Hessian terms, directional derivative, and a tangent plane approximation at a chosen point.
No. It uses numerical finite differences. That makes it practical for many smooth functions, but very sharp functions or poor step sizes can reduce accuracy.
You can use x, y, and z. If your function depends on only two variables, set z to any constant and dz to zero.
A small value like 0.0001 often works well. If the function is noisy or extremely large, test a few nearby step sizes and compare stability.
It measures how fast the function changes in the chosen direction. The calculator normalizes your direction vector before computing this rate.
Mixed partials reveal interaction between variables. They help describe local curvature and are key parts of the Hessian matrix.
If the gradient magnitude is near zero, the point may be stationary. You should then inspect second derivatives or the Hessian for local classification.
Yes. After calculation, use the CSV or PDF buttons to download a compact summary of the computed values and matrix entries.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.