Fractional Derivatives

Question

If we define the (forward) difference operator as $$\Delta f(x)=f(x+\Delta x)-f(x)$$ we can break it up using the "shift" operator $E\,f(x)=f(x+\Delta x)$ and the "identity" $1\,f(x)=f(x)$. Then $\Delta =E-1$ so we can write the derivative as $$Df(x)=\lim_{\Delta x\to 0}\frac{\Delta f(x)}{\Delta x}=\lim_{\Delta x\to 0}\frac{(E-1)f(x)}{\Delta x}$$ In general the $n$'th derivative is $$D^nf(x)=\lim_{\Delta x\to 0}\frac{\Delta^n f(x)}{\Delta x^n}=\lim_{\Delta x\to 0}\frac{(E-1)^n f(x)}{\Delta x^n}$$ I tried defining the fractional derivative as $$\lim_{\Delta x\to 0}\Bigg[{1\over \Delta x^n}\left(1+\binom{n}{1}(E-2)+\binom{n}{2}(E-2)^2+\cdots\right)f(x)\Bigg]$$ where I have expanded $(E-1)^n$ into a power series. Unfortunately, I haven't the slightest clue how to evaluate this :( For example, using this definition, what would $D^{1/2}(x)$ be?