TITLE: Local fractional derivative that doesn't vanish on differentiable functions
QUESTION [5 upvotes]: Riemann-Liouville fractional derivative is a nonlocal fractional derivative that doesn't vanish in general on differentiable functions. Kolwankar-Gangal 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
$$
   D^{\alpha} x^{n \alpha} = \frac{\Gamma(n\alpha+1)}{\Gamma((n-1)\alpha+1)} x^{(n-1)\alpha}
$$
holds for any $x > 0$?

REPLY [6 votes]: By the theorem of Peetre, a linear operator $C^\infty(\mathbb R)\to  C^\infty(\mathbb R)$ which is local (=support non-increasing) is a differential operator (so involves only integer derivatives).