In calculus, whole-number derivatives and integrals of functions are simple to interpret geometrically: the derivative is the rate of change of a function, whereas the integral is the area between a function and the x-axis. In a letter to L’Hospital in 1695, Leibniz raised the possibility of generalizing the operation of differentiation to non-integer orders. What would a half-derivative or half-integral mean? Despite this early interest in fractional calculus, it was not until 1974 that fractional calculus became the object of specialized mathematics conferences and texts. In recent years, applications to chemistry and physics have increased the interest in fractional calculus. Within these applications, competing definitions for the fractional integrals have been proposed: the Riemann-Liouville, Weyl, and Caputo fractional derivatives being the three most prominent. Unlike with ordinary differentiation, the fractional derivative of a function  depends on which definition for the fractional derivative is chosen; in other words, the fractional derivative of a function is not unique.

The focus of this research was on analytically finding and graphically showing the continuous interpolation between elementary real-valued functions and their first derivatives, as alpha varies continuously from 0 to 1 for the Caputo fractional derivatives. The elementary functions under study include constant functions, rational powers of x, and exponential functions. The contribution of this research was two-fold. First, it provided a record of the derivation of the Caputo fractional derivatives for elementary real-valued functions. Second, it provided the graphical interpretation of these derivatives by presenting the constant interpolation between the function and its first derivative. This research also addresses a gap in the mathematics literature identified by Machado: there is no elaboration of the simple concepts from fractional calculus that are approachable to undergraduates and could easily be incorporated into existing third-semester calculus courses. This research provides such concepts and examples.