Analytic slope, central-difference slope, and absolute error by xUpdates with every input
Math & Statistics
Derivative Value Table Calculator
Build a live derivative table for f(x)=A sin(kx)+Bx squared+Cx+D, compare the analytic derivative with central differences, and expose truncation-versus-roundoff behavior across step sizes.
Maximum absolute error -
RMS derivative error -
Smallest-error row -
Largest-error row -
Mean analytic slope -
Direction matches -
Difference step -
Rows generated -
NUMERICAL DIFFERENTIATION ERROR
Analytic slope, central-difference slope, and absolute error by x
The upper comparison uses one scale for both slope estimates; the lower strip magnifies their absolute difference.
DERIVATIVE VALUE TABLE
Exact derivative and finite-difference estimate at every x row
The same h is applied symmetrically around each x, including the table endpoints.
| x | f(x) | Analytic f'(x) | Central difference | Signed error | Relative error |
|---|
TABLE SETUP
Choose x coverage and h independently
- Define the sine-plus-quadratic function.
- Set table start below table end.
- Choose enough rows to resolve the interval.
- Set h relative to the function scale.
- Compare absolute and relative error before trusting digits.
STEP-SIZE TRADEOFF
Smaller h reduces truncation error only until subtraction loses precision
For smooth functions, central differences generally improve quadratically as h shrinks from a coarse value. But f(x+h) and f(x-h) become nearly equal, so floating-point subtraction eventually loses significant digits.
Relative error can look enormous near a true zero derivative. Absolute error and direction agreement remain necessary companion checks.
Subject fundamentals
Five ideas that control this calculation
The analytic derivative is the reference
The sine term follows the chain rule, the quadratic follows the power rule, and the constant disappears before any table row is evaluated.
The numerical estimate is symmetric
The central difference uses f(x+h) and f(x-h), balancing the two sides of x and canceling the leading one-sided error term.
h controls two error sources
A coarse step increases truncation error, while an extremely small step magnifies subtraction and floating-point roundoff.
The grid is inclusive
The requested row count spaces x-values evenly from the entered start through the entered end, including both endpoints.
Error needs more than one summary
Maximum error finds the worst row, RMS error describes the table overall, and sign agreement checks whether directional interpretation survives.
CENTRAL-DIFFERENCE COMPARISON
Use the symbolic derivative as a reference for a numerical approximation
The central difference samples equally on both sides of x. Its leading truncation error is proportional to h squared for a smooth function, while excessively tiny h can amplify floating-point cancellation.
Detailed calculation process and general formulas
f(x)=A sin(kx)+Bx^2+Cx+Df'(x)=Ak cos(kx)+2Bx+CD_h f(x)=[f(x+h)-f(x-h)]/(2h)error(x)=D_h f(x)-f'(x)Symbols, meanings, and units
- A,k
- sine amplitude and angular frequencymodel-dependent
- B,C,D
- polynomial coefficientsmodel-dependent
- h
- central-difference half-stepx units
- D_h f
- central-difference derivative estimatey units per x unit
- error
- numerical estimate minus analytic derivativey units per x unit
NUMERICAL QUALITY
Three checks before accepting a derivative table
Accuracy is not represented by one percentage alone.
Worst point
-Maximum absolute error identifies the least accurate row.
Whole-table fit
-RMS error summarizes the overall approximation.
Direction consistency
-Sign matches verify increasing/decreasing interpretation.
Decision takeaway: Test more than one h when the table supports a consequential numerical decision.
REPRODUCIBILITY NOTES
Record these with a finite-difference result
- Function expression
- Input units
- Chosen h
- Floating-point precision
- x grid
- Reference derivative
Applied decisions
Where the comparison exposes numerical behavior
Coarse step on an oscillatory term
k is increased while h remains large.
What the result clarifies: The error strip grows because the local wave curvature is under-resolved.
Near-zero analytic slope
A row lands near a turning point.
What the result clarifies: Relative error becomes unstable even though absolute error can remain small.
Worked default scenario
Current-input substitution and reconciliation
Key terms
Glossary for interpreting the result
- Central difference
- The estimate [f(x+h)-f(x-h)]/(2h).
- Step size
- The positive offset h used on each side of a table x-value.
- Analytic derivative
- The exact symbolic derivative used as the comparison reference.
- Signed error
- Numerical derivative minus analytic derivative.
- RMS error
- The square root of the mean of squared row errors.
- Direction match
- Agreement between the signs of numerical and analytic slopes.
Method references
References for this calculator's specific method
Scope and limitations
The analytic derivative is exact for the displayed function family within floating-point arithmetic. Numerical-difference behavior may differ for noisy data, discontinuities, black-box simulations, or lower-precision systems.
Derivative Value Table Calculator | Analytic vs Central Difference FAQ
Why use a central difference instead of forward difference?
For smooth functions it has second-order truncation error and usually provides better local symmetry.
Is a smaller h always better?
No. Very small h can amplify floating-point cancellation and measurement noise.
Why is relative error large near zero slope?
The denominator is close to zero, so a small absolute discrepancy becomes a large percentage.
Does D affect the derivative?
No. The constant cancels analytically and in the symmetric difference.