As with the counting numbers which by subtraction gave us the integers and by ratios became our rational numbers, mathematicians explore possible fractionation of countable processes: In particular here the derivative is defined as such a complete, unit, of process, and successive derivatives iterate the count: Negative derivatives indicate the inverse operation called, integration, of which the derivative is the original function; and fractional derivatives might indicate processes that can be multiply iterated to achieve the equivalent of one, full, unit of derivation (derivative).
For example, xδi c xn = c (n!/n-i!) xn-i . A fractional-derivative here, has fractional values for the derivative iterator, i ,- which works, upon inspection,- and in implementation, by the Sterling approximation to n! ...
Fractional derivatives have obvious utility, for example in hand-controlled motions, rovot finger-wiggle-pad control of computer-cursors, automobile cruise-drive controls,- where typically the center finger-range is absolute positional, the midway-out range is absolute velocity, and the edge is near or absolute acceleration: so that the user has continuous fine-to-fast-to-free control, without having to lift a finger back stroke the center for more. (Of course in practical design the temporal motion would be included in the operator equation, to avoid erratic detuned control on fingerprint grip-slip.)