A Simple Engineering Technique to Estimate the First Derivative of an Empirical Function

TL;DR

This paper introduces a heuristic 5-point approximation formula for estimating the first derivative of empirical functions measured at unequally spaced points, accounting for measurement errors and smoothing effects.

Contribution

It proposes a novel, simple formula with fixed optimal weights for derivative estimation that improves accuracy over classical methods, applicable across various empirical data scenarios.

Findings

Optimal weight coefficients are consistent across different functions.

The method significantly increases derivative estimation accuracy with small measurement errors.

The formula is effective for diverse applications like thermal processes and material property analysis.

Abstract

A heuristic formula for 5-point approximation of the first derivative of an unknown function whose values are measured with an error at unequally spaced points is proposed. The derivative at a given point is calculated using the effective increments of the function and the argument, taking into account the different weight coefficients for near and far measurement points. Simulation modeling on test functions with known derivatives is applied to determine rational values of weight coefficients. Simulation results are described in detail on two test functions, one of which simulates the process of water cooling of a hot steel sheet, the second is a complex oscillatory process with variable frequency and amplitude. It was found that the optimal values of weight coefficients remain approximately the same for essentially different functions, which allows us to recommend the same formula for all cases. In contrast to the classical methods of numerical differentiation of functions tabulated at unequally spaced nodes, the proposed formula simultaneously takes into account the smoothing of empirical data. It is shown that this significantly increases the accuracy of the numerical derivative estimate even in cases where the random error of the function measurement is a very small value, from 1%. The formula obtained is recommended for use in solving any problems requiring the estimation of the derivative of an empirical function, including the calculation of the stress-strain state of metal, the description of thermal processes, and the determination of the properties of materials.