Leibniz wrote, "Thus it follows that d^1/2x will be equal to x (dx:x)^(1/2), an apparent paradox, from which one day useful consequences will be drawn." G.W. Leibniz, Letter from Hanover, Germany, September 30, 1695 to G.A. L'Hospital.

The fractional derivative can be computed in a number of ways, one such way is in the frequency domain. Denoting the Fourier transform of the function f(x) as F(w), it is straight-forward to show that the Fourier transform of the n^th-order derivative, f⁽ⁿ⁾(x), is (jw)ⁿF(w), for any integer order n. Of course, there is no reason why n must be an integer, n can be any real (or complex) number - hence the fractional derivative. If n<0, then we have fractional integration.

Illustrated below are the zeroth through fourth-order normalized derivatives of a Gaussian in steps of 0.5. The last panel illustrates a continuum of derivatives from first to fourth (top to bottom) "stacked" on top of one another.



Matlab routine for computing fractional derivatives.