Compute first and second derivatives from tabulated measurements. Choose stencil size, target point, and precision. See results instantly with formulas, tables, and downloadable reports.
Enter experimental or simulated x-y data. The tool sorts x values automatically, builds finite-difference weights, and estimates the requested derivative at the target point.
This sample represents position versus time for uniformly accelerated motion. At time 2 s, the first derivative is velocity and the second derivative is acceleration.
| Time (s) | Position (m) |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
For a selected set of nearby points, the calculator builds finite-difference weights that reproduce the target derivative through a local Taylor-series match.
f^(m)(x₀) ≈ Σ [ wᵢ · f(xᵢ) ]
The weights are chosen so that polynomial terms up to the stencil degree behave correctly at the target point. This lets the calculator work with centered, forward, backward, uniform, and non-uniform point patterns.
For equally spaced central data, common special cases include:
First derivative: f'(x₀) ≈ [f(x₀+h) - f(x₀-h)] / (2h)
Second derivative: f''(x₀) ≈ [f(x₀+h) - 2f(x₀) + f(x₀-h)] / h²
In physics, the first derivative often represents rate, such as velocity from position. The second derivative often represents curvature or acceleration.
It estimates how fast a measured quantity changes with respect to another variable. In physics, that often means velocity from position data, acceleration from velocity data, or local slope from experimental observations.
Yes. The calculator generates finite-difference weights directly from your selected nodes, so it can handle non-uniform spacing. Very irregular spacing may still reduce stability, so moderate point distribution is usually better.
Smaller stencils react faster and need fewer points. Larger stencils often smooth noise and improve local polynomial accuracy, but they can blur rapid changes. A 5-point stencil is a practical starting choice for many datasets.
Use forward selection near the beginning of sampled data and backward selection near the end. Centered selection is usually best for interior points because it balances information from both sides of the target.
It is the absolute difference between the chosen stencil result and a smaller stencil result. It is not a strict error bound, but it gives a useful practical check on local sensitivity.
Yes. Differentiation amplifies noise, especially for higher-order derivatives. If your data are noisy, consider smoother measurements, moderate stencil sizes, or pre-processing before relying on the derivative for interpretation.
Yes. If position is your y variable and time is x, the second derivative gives acceleration units such as m/s². The first derivative would give velocity in units such as m/s.
The derivative divides the y unit by the x unit raised to the derivative order. For example, meters over seconds gives m/s, while meters over seconds squared gives m/s².
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.