TITLE: $\frac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}}$, what does it mean?
QUESTION [5 upvotes]: I would like to know what these fractional exponents means in a derivative $\frac{d}{dx}$ operator. Like, I've seen $\frac{d^2}{dx^2}$ but I don't know what $\frac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}}$ means.

REPLY [4 votes]: What you have is what is usually referred to as a semiderivative. I would say that this notation is a bit incomplete, since one also has to account for a lower limit, in the case of differintegrals whose orders are not nonnegative integers.
In particular, letting ${}_a D_x^{1/2}f(x)$ be the semiderivative, one possible definition is the Riemann-Liouville form,
$${}_a D_x^{1/2}f(x)=\frac{f(a)}{\sqrt{\pi(x-a)}}+\frac1{\sqrt\pi}\int_a^x \frac{f^\prime(t)}{\sqrt{x-t}}\mathrm dt$$
The semiderivative operator can be thought of this way: this is the operator that, when applied to a function twice, gives the ordinary (first order) derivative.
If you want to learn more about this, I would suggest picking up Oldham/Spanier's The Fractional Calculus.