RiemannLiouville fractional derivative is a nonlocal fractional derivative that doesn't vanish in general on differentiable functions. KolwankarGangal fractional derivative is local but vanishes on any differentiable function. Is there some local fractional derivative that doesn't vanish on differentiable functions in general and for which $$ D^{\alpha} x^{n \alpha} = \frac{\Gamma(n\alpha+1)}{\Gamma((n1)\alpha+1)} x^{(n1)\alpha} $$ holds for any $x > 0$?

$\begingroup$ I'm not sure, but maybe you could investigate the Yang local fractional derivative? $\endgroup$ – Petern Jan 25 '13 at 17:04
By the theorem of Peetre, a linear operator $C^\infty(\mathbb R)\to C^\infty(\mathbb R)$ which is local (=support nonincreasing) is a differential operator (so involves only integer derivatives).
Not a translationinvariant one!
Indeed, we would have:
$$ \frac{ \Gamma( 2) } {\Gamma(\frac{3}{2}) } \sqrt{x+1} = D^{1/2} (x+1)= D^{1/2}x + D^{1/2} 1 = \frac{ \Gamma( 2) } {\Gamma(\frac{3}{2}) } \sqrt{x} +\frac{\Gamma(1)} {\Gamma(\frac{1}{2} ) } \frac{1}{\sqrt{x}} $$
which is obviously false.