Numerical Differentiation Calculator | Forward, Central & Richardson
Approximate the derivative of any function f(x) at a given point using forward difference, backward difference, central difference (O(h²) accurate), and Richardson extrapolation (O(h⁴) accurate). Also computes the second derivative via the central second-difference formula and shows the truncation error for each method.
Use: sin, cos, tan, exp, log, sqrt, abs, pow(a,b), PI, E — and standard operators + - * / ( )
What Is the Numerical Differentiation Calculator | Forward, Central & Richardson?
Numerical differentiation approximates derivatives using only function values at nearby points, without requiring an analytic formula. Forward difference uses f(x) and f(x+h) with O(h) truncation error. Backward difference uses f(x−h) and f(x) — same accuracy. Central difference uses f(x+h) and f(x−h) symmetrically and achieves O(h²) accuracy by canceling the first-order error terms. Richardson extrapolation applies central difference at both h and h/2, then combines them as [4C(h/2)−C(h)]/3 to cancel the h² error, achieving O(h⁴) accuracy with only twice the central-difference cost. The second derivative uses [f(x+h)−2f(x)+f(x−h)]/h², also O(h²).
Formula
f′(x) ≈ [f(x+h)−f(x−h)] / (2h) | Richardson: [4·C(h/2)−C(h)] / 3 | f″(x) ≈ [f(x+h)−2f(x)+f(x−h)] / h²
How to Use
- 1
Enter f(x) using standard math notation: sin(x), cos(x), exp(x), log(x), sqrt(x), x*x, 1/x, etc.
- 2
Set x — the point where you want to evaluate the derivative
- 3
Set h — the step size (typically 0.001 to 0.0001; avoid values smaller than 10⁻¹²)
- 4
Click a preset to load a classic function with a known exact derivative
- 5
Click "Differentiate Numerically" — all four first-derivative methods are shown
- 6
Read the error column (relative to Richardson) to see how each method compares
- 7
The second derivative f″(x) appears in the summary card above the table
Enter f(x) as a math expression, set evaluation point x and step size h, then click Differentiate Numerically.
Example Calculation
f(x) = sin(x) at x = π/4, h = 0.001. Exact f′(π/4) = cos(π/4) ≈ 0.70710678. Forward: 0.70675 (error 5×10⁻⁴). Central: 0.70710 (error 2×10⁻⁷). Richardson: 0.70710678 (error < 10⁻¹²). Richardson is 5,000× more accurate than forward difference using the same function evaluations.
Understanding Numerical Differentiation | Forward, Central & Richardson
Method Comparison
| Method | Formula | Function evals | Truncation error | Best h |
|---|---|---|---|---|
| Forward diff. | [f(x+h)−f(x)] / h | 2 | O(h) | ~10⁻⁵ |
| Backward diff. | [f(x)−f(x−h)] / h | 2 | O(h) | ~10⁻⁵ |
| Central diff. | [f(x+h)−f(x−h)] / 2h | 2 | O(h²) | ~10⁻⁴ |
| Richardson | [4·C(h/2)−C(h)] / 3 | 4 | O(h⁴) | ~10⁻³ |
| 2nd deriv. | [f(x+h)−2f(x)+f(x−h)] / h² | 3 | O(h²) | ~10⁻⁴ |
Exact Derivatives for Verification
| f(x) | f′(x) exact | f″(x) exact | Example: f′(1) |
|---|---|---|---|
| sin(x) | cos(x) | −sin(x) | 0.54030 |
| cos(x) | −sin(x) | −cos(x) | −0.84147 |
| exp(x) | exp(x) | exp(x) | 2.71828 |
| log(x) | 1/x | −1/x² | 1.00000 |
| x² | 2x | 2 | 2.00000 |
| x³ | 3x² | 6x | 3.00000 |
| 1/x | −1/x² | 2/x³ | −1.00000 |
| sqrt(x) | 1/(2√x) | −1/(4x^(3/2)) | 0.50000 |
The Step-Size Dilemma
- ›Truncation error decreases as h → 0: smaller h means the finite difference more closely matches the true derivative.
- ›Roundoff error increases as h → 0: f(x+h) − f(x) becomes the difference of two nearly equal numbers, losing significant digits in floating-point arithmetic.
- ›Optimal h balances these two effects. For double precision (ε ≈ 10⁻¹⁶): forward diff. optimal h ≈ √ε ≈ 10⁻⁸, central diff. optimal h ≈ ε^(1/3) ≈ 10⁻⁵, Richardson optimal h ≈ ε^(1/5) ≈ 10⁻³.
- ›Richardson extrapolation shifts the optimal h to a larger value (~10⁻³) while achieving better accuracy — a practical advantage for most purposes.
Frequently Asked Questions
Why is Richardson extrapolation more accurate?
It applies central difference at two step sizes h and h/2, then combines them as [4·C(h/2) − C(h)] / 3. This algebraically cancels the dominant h² error term, leaving an O(h⁴) remainder — four orders of magnitude smaller per halving of h.
Why not use an infinitely small h?
Very small h causes catastrophic cancellation: f(x+h) − f(x) is the difference of two nearly equal floating-point numbers, losing significant digits. The optimal h balances truncation error (decreases with h) against roundoff error (increases as h → 0).
What functions can I use?
sin, cos, tan, asin, acos, atan, exp, log (natural), log10, log2, sqrt, abs, pow(a,b), ceil, floor, round, and standard operators +, -, *, /, (, ).
When would I use forward instead of central difference?
When you can only evaluate f on one side of x — for example, at a boundary of a domain where x−h is invalid. Central difference is almost always preferred when both sides are accessible.
What is the second derivative formula used here?
The central second-difference: [f(x+h) − 2f(x) + f(x−h)] / h². It has O(h²) accuracy. The second derivative measures the curvature (rate of change of slope) at x.
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