Five measured points, two secants, and one center-slope estimateLive current inputs
Math & Statistics
Derivative Approximation Calculator
Estimate a local derivative from five equally spaced observations, compare three-point and five-point stencils, and expose asymmetry, curvature, and data-consistency limits.
Five-point derivative -
Three-point derivative -
Forward local slope -
Backward local slope -
Second derivative estimate -
One-sided slope spread -
Five-vs-three gap -
Local direction -
LOCAL SAMPLE GEOMETRY
Five measured points, two secants, and one center-slope estimate
The live stencil keeps the observations visible, draws the left and right one-step secants, and overlays the five-point tangent without inventing a global fitted curve.
WEIGHTED STENCIL LEDGER
How each observation contributes to the five-point derivative
The coefficient pattern is antisymmetric: the center value carries zero derivative weight while outer and inner pairs oppose each other.
| Sample | x-coordinate | Observed y | Derivative weight | Weighted contribution |
|---|
DATA ENTRY
Preserve equal spacing and raw measurement order
- Set x0 to the central observation.
- Enter the exact constant spacing between samples.
- Enter measured y values from left to right without sorting by magnitude.
- Retain measurement precision rather than rounding every point to a different resolution.
- Use the slope spread as a local diagnostic, not a universal uncertainty interval.
WHAT THE STENCIL KNOWS
The estimate is local and does not assume a global function family
A five-point stencil is useful when observations are evenly spaced and the response is sufficiently smooth over four intervals. It does not fit or extrapolate a polynomial beyond the sample window.
The difference between forward and backward slopes contains real curvature plus noise. A large spread should prompt inspection of the measurements and the choice of h.
FINITE-DIFFERENCE STENCIL
Use symmetric evidence to suppress low-order truncation error
The five-point formula combines two symmetric distances. Comparing it with the three-point and one-sided slopes shows how strongly the estimate depends on curvature and local asymmetry.
Detailed calculation process and general formulas
D5 = [y(-2)-8y(-1)+8y(+1)-y(+2)]/(12h)D3 = [y(+1)-y(-1)]/(2h)D+ = [y(+1)-y(0)]/hD- = [y(0)-y(-1)]/hf''(x0) ≈ [y(-1)-2y(0)+y(+1)]/h^2Symbols, meanings, and units
- h
- equal spacing between adjacent observationsx-unit
- y(k)
- observed response at x0+khy-unit
- D5
- five-point center derivativey-unit/x-unit
- D3
- three-point center derivativey-unit/x-unit
- D+
- right one-step secant slopey-unit/x-unit
- D-
- left one-step secant slopey-unit/x-unit
FIELD INTERPRETATION
Read the derivative together with two shape diagnostics
One slope number cannot reveal whether the neighborhood is straight, curved, or noisy.
Best center estimate
-The five-point stencil uses all four off-center observations.
Local bend
-The second difference approximates curvature around x0.
Left-right disagreement
-The one-sided spread flags asymmetry or noise.
Decision takeaway: Report the stencil, spacing, and local disagreement whenever an approximate derivative is used in a decision.
MEASUREMENT NOTES
Record context the five numbers cannot preserve
- Sensor resolution and calibration
- Timestamp or spatial station of each sample
- Whether h is exact or rounded
- Known discontinuities or regime changes
- Any smoothing applied before entry
Applied decisions
Where a five-point derivative is more useful than a guessed curve
Temperature gradient
Five equally spaced probes straddle a wall section.
What the result clarifies: The center slope is reported alongside curvature and left-right disagreement.
Demand response
Five price points surround the current price.
What the result clarifies: The local marginal estimate avoids claiming a global demand model from a narrow experiment.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
The five-point formula assumes equal spacing and a smooth response. It is not appropriate across jumps, corners, phase changes, or reordered samples. Measurement uncertainty can dominate the formal truncation advantage.
Derivative Approximation Calculator | Five-Point Sample Stencil FAQ
Why is the center value absent from D5?
The derivative weights are antisymmetric, so the center carries zero weight for an odd local derivative.
Is five-point always more accurate than three-point?
Only for sufficiently smooth, accurately measured, equally spaced data. Extra points can also carry extra noise.
Can h be negative?
Enter a positive spacing and preserve left-to-right sample order.
What does a large asymmetry mean?
It can reflect curvature, noise, a poor center location, or a nearby regime change.