Central-difference error should descend quadratically before rounding dominatesLive current inputs
Math & Statistics
Derivative Verification Calculator
Audit a claimed derivative of a quartic polynomial at one point by comparing the analytic rule, central differences at successively smaller steps, and an entered acceptance tolerance.
Analytic derivative -
Claim absolute error -
Finest central difference -
Finest numerical error -
Observed convergence order -
Claim verdict -
Constant-term check -
Numerical stability -
SLOPE CONVERGENCE AUDIT
Central-difference error should descend quadratically before rounding dominates
The chart plots absolute error against step size on log scales, marks the claim separately, and exposes whether halving h produces the expected second-order pattern.
INDEPENDENT CHECK REGISTER
Five step sizes, five slope estimates, one analytic reference
Each row is recomputed directly from function values; it does not reuse the analytic derivative except to report the audit error.
| Step h | f(x0-h) | f(x0+h) | Central difference | Absolute error | Error ratio |
|---|
AUDIT SETUP
Verify the same function, point, and units used by the original claim
- Enter every polynomial coefficient, including zero coefficients.
- Enter the exact x-coordinate used by the claimed derivative.
- Record the claim before looking at the analytic answer.
- Choose a starting h that is small relative to the x scale but not near machine precision.
- Interpret convergence across rows instead of trusting one numerical estimate.
WHY TWO METHODS
Agreement is stronger when the checks fail in different ways
Symbolic differentiation can be undermined by a copied coefficient or exponent. Numerical differentiation can be undermined by an unsuitable step. Their agreement across a converging sequence is more informative than either answer alone.
The constant coefficient must disappear from the analytic derivative. The page reports that invariance explicitly because retaining e is a common rule error.
VERIFICATION METHOD
Differentiate symbolically, then challenge the result with sampled function values
The analytic derivative supplies the reference. A symmetric finite difference cancels the first-order truncation term, so its error normally falls by about four when h is halved.
Detailed calculation process and general formulas
f(x) = ax^4 + bx^3 + cx^2 + dx + ef'(x0) = 4ax0^3 + 3bx0^2 + 2cx0 + dD_h = [f(x0+h)-f(x0-h)]/(2h)E_h = |D_h-f'(x0)|p = log(E_h/E_(h/2))/log(2)Symbols, meanings, and units
- x0
- point where the derivative is checkedx-unit
- h
- symmetric sampling stepx-unit
- D_h
- central-difference slopef-unit/x-unit
- E_h
- absolute numerical errorf-unit/x-unit
- p
- observed error-convergence orderdimensionless
- tol
- maximum accepted claim errorf-unit/x-unit
REVIEW SIGNALS
Three distinct reasons an audit can fail
A red claim is not automatically a bad analytic formula, and a noisy sequence is not automatically a bad claim.
Claim mismatch
-The entered claim lies outside the allowed analytic tolerance.
Discretization mismatch
-The finite-difference estimate has not yet reached the analytic reference.
Step instability
-Error stops declining predictably, indicating scale or rounding limits.
Decision takeaway: Accept the claim only after the analytic rule and a stable independent sequence point to the same slope.
COMMON FAILURE PATTERNS
What the error trajectory reveals
- Flat high error often means the function or x0 was copied incorrectly.
- A fourfold error decrease after halving h is consistent with a central difference.
- A sudden last-row increase can indicate cancellation and should not be hidden.
- Changing e should not change any derivative estimate except the raw sampled function values.
Applied decisions
Derivative claims worth auditing
Spreadsheet formula review
A forecast workbook reports a marginal response from a quartic fit.
What the result clarifies: The convergence register distinguishes a copied derivative error from harmless display rounding.
Classroom solution check
Two solutions disagree only in the constant and linear terms.
What the result clarifies: The constant gate and independent slopes show which term can affect the derivative.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
Finite differences are verification tools, not exact symbolic proofs. Very large h introduces truncation error; extremely small h can amplify floating-point cancellation. Scale variables before using this page for ill-conditioned scientific models.
Derivative Verification Calculator | Analytic and Numerical Slope Audit FAQ
Why does the table halve h?
A controlled step sequence reveals whether the numerical method is converging at its expected rate.
Should the error always decrease?
Not indefinitely. At very small steps, subtractive cancellation can reverse the trend.
Why is the constant checked separately?
A true derivative of a constant is zero, so any retained constant contribution is a rule error.
Does a passing claim prove the polynomial model is correct?
No. It verifies the derivative of the entered polynomial, not whether that polynomial describes the real system.