Experimental Mathematics: Discovering The Half-Derivative from First Principles

Abstract

This expository paper grew out of two lectures in Miller's Math 409: The little Questions (Williams College, Fall 2026). We explore fractional derivatives by asking what a ``half-derivative'' might mean if applying it twice is to recover an ordinary derivative. Linearity and a natural guess for the half-derivative of a power function lead to a functional equation related to the classical factorial recurrence. Choosing the familiar Gamma-function extension gives the classical formula \[ \mathscr{D}^{1/2}(x^n) \ =\ \frac{Γ(n+1)}{Γ(n+\tfrac12)}x^{n-1/2}. \] We derive $Γ(n+1)=n!$, evaluate $Γ(1/2)=\sqrtπ$ using the Gaussian integral, and verify the half-derivative formula with an elementary integral. Along the way, we explore how the Gamma function generalizes factorials and binomial coefficients beyond their classical domains, encounter the double factorial, see why the usual product rule fails, and examine the half-derivative of $e^x$. We also explain why the original requirements do not determine a unique operator. The treatment emphasizes motivation, the benefits of experimentation through accessible calculations in test cases, rather than the general theory of fractional calculus.