How can I define the fractional derivative of the Delta function? 

I mean $D^{\alpha}= \frac{d^{\alpha}}{dx^{\alpha}} $ where $\alpha$ can be any real number, then if we define $D^{\alpha} \delta (x) $ how can we define it in the sense of distribution?

Applying formal integration by parts $ \alpha $ times I guess that

$$ 
\int\_{-\infty}^{+\infty}D^{\alpha}\delta (x) g(x)dx= (-1)^{[ \alpha]}\int\_{-\infty}^{+\infty}D^{\alpha}g(x)\delta(x)dx= (-1)^{[ \alpha]}D^{\alpha}g(0)
$$

for any test function $ g(x) $.