Estimate derivatives directly from three measured observations without entering an analytic function. Enter y0, y1 and y2 in increasing position order, with the same positive spacing h between consecutive positions: x1 = x0 + h and x2 = x0 + 2h. You do not need to enter x0.
For exact samples of a smooth function, the first-derivative estimates have O(h²) truncation error when the third derivative is continuous and bounded nearby. The midpoint second derivative has O(h²) error when the fourth derivative is continuous and bounded. Here, second-order accuracy describes error scaling, not derivative order.
Equal spacing is essential; do not substitute an average spacing for irregular observations. Noise and rounding are amplified by division by h, and more strongly by h² for the second derivative. Smaller spacing does not automatically improve measured-data estimates. See Berkeley’s numerical differentiation notes.
| Estimate | Result |
|---|---|
| First derivative at x0 = 0 | 0 |
| First derivative at x1 = 1 | 2 |
| First derivative at x2 = 2 | 4 |
| Second derivative at x1 = 1 | 2 |
All inputs use dimensionless numeric values; no unit conversions are performed. For measured quantities, use consistent units for all three observations and one consistent position unit for h. Interpret first derivatives as observation units per position unit and second derivatives as observation units per squared position unit.
Spacing must be strictly positive. Zero spacing makes the results unavailable because it causes division by zero. Very large or small values can exceed the numerical range, and dependent results may also become unavailable. Unavailable results are not zero. Displayed decimal places do not indicate measurement accuracy.
Use this calculator for three equally spaced observations. Use the expression-based derivative calculator when you have a function expression. The two-point slope calculator calculates a secant slope between two points. Richardson extrapolation combines already-computed estimates at different step sizes.