A derivative is equivalent to a highpass filtering with a slope of +20 dB/decade (or about +6 dB/octave), and an integral a lowpass filter of -20 dB/dec. So if you filter something by +10 dB/dec, you get half a derivative.
To be more specific, if we write the Fourier transform of a function as FT(x(t)), then FT(d/dt x(t)) = j2pifFT(x(t)).
In fact, this equality holds for higher order derivatives:
FT(d^n/dt^n x(t)) = (j2pif)^n FT(x(t))
Where d^n/dt^n is the nth derivative.
We naturally extend this definition to "1/2 derivatives" the same way we often extend integer valued functions to take rational arguments: we plug in a rational and see what happens:
FT(d^(1/2)/dt^(1/2) x(t)) = (j2pif)^(1/2) FT(x(t)) = sqrt(j2pif) FT(x(t))
Then we take the inverse Fourier transform to find the half-derivative, which is what we were originally looking for:
d^(1/2)/dt^(1/2) x(t) = FT^-1 (sqrt(j2pif) FT(x(t)))