Abstract

This paper proposes a novel generalization of the classical derivative by replacing the concept of a tangent line with a local arc approximation. Instead of identifying the instantaneous rate of change as the slope of a linear tangent, we define the local behavior of a function by fitting a circular arc of radius r and center a to a neighborhood of the point of interest. This "arc-based flow" captures not only the direction of change (first-order behavior) but also the intrinsic curvature of the function (second-order behavior) through a unified geometric framework. We formalize this construction, show its convergence to classical derivatives as the arc length tends to zero, and discuss computational methods and theoretical implications.