Calculator Form
Use function mode for a formula such as sin(x), exp(x), or x^3. Use table mode for equally spaced x-values and matching y-values.
Formula Used
First derivative
Function form: f'(x) ≈ [f(x+h) - f(x-h)] / (2h)
Table form: f'(xᵢ) ≈ [yᵢ₊₁ - yᵢ₋₁] / (2h)
This estimate is second-order accurate for smooth data, so the truncation error is typically proportional to h².
Second derivative
Function form: f''(x) ≈ [f(x+h) - 2f(x) + f(x-h)] / h²
Table form: f''(xᵢ) ≈ [yᵢ₊₁ - 2yᵢ + yᵢ₋₁] / h²
Second derivatives are sensitive to noise, so a smaller h is not always better when values are rounded or measured.
How to Use This Calculator
- Select Function expression to evaluate a formula directly, or choose Equally spaced table for tabulated data.
- Pick the derivative order. Use first derivative for slope, or second derivative for curvature.
- For function mode, enter f(x), the target x, and the step size h.
- For table mode, enter equally spaced x-values, matching y-values, and a target x that exists in the list.
- Optionally enter an exact derivative value to compute absolute and relative error.
- Click Calculate Central Difference to display the result above the form.
- Review the stencil values, comparison table, and Plotly graph.
- Use the CSV and PDF buttons to export your calculation summary.
Example Data Table
This sample uses values from y = sin(x) near x = 1.0 with spacing h = 0.1. It gives a first-derivative estimate close to cos(1).
| x | y = sin(x) |
|---|---|
| 0.8 | 0.717356 |
| 0.9 | 0.783327 |
| 1.0 | 0.841471 |
| 1.1 | 0.891207 |
| 1.2 | 0.932039 |
Frequently Asked Questions
1) What does the central difference method estimate?
It estimates derivatives using values on both sides of a target point. That balanced stencil usually gives better accuracy than one-sided estimates when the function or table is smooth.
2) Why is central difference often better than forward or backward difference?
For smooth data, the basic central first-derivative formula has second-order accuracy. Forward and backward first differences are usually only first-order accurate with the same step size.
3) Must table data be equally spaced?
Yes. The formulas implemented here assume uniform spacing. If the x-values are uneven, you should use interpolation or divided-difference methods designed for nonuniform grids.
4) Why can’t the target x be the first or last table value?
Central difference needs one point on each side of the target. Endpoints only have data on one side, so forward or backward difference would be required instead.
5) How should I choose h?
A moderate h often works best. If h is too large, truncation error grows. If h is too small, rounding or measurement noise can dominate the estimate.
6) What is the purpose of the exact derivative field?
It lets the calculator report absolute and relative error. That is useful for checking step-size choices, comparing methods, and validating classroom examples or benchmark problems.
7) Which function expressions are supported?
You can use x, numbers, parentheses, operators, and common functions such as sin, cos, tan, exp, log, sqrt, and abs. Use explicit multiplication like 3*x.
8) Can I use this calculator with noisy measurements?
Yes, but interpret results carefully. Second derivatives are especially noise-sensitive. Smoother data, moderate spacing, and comparison with known trends can improve reliability.