Of course, I'm not really familiar with all fractional derivative methods, but is it a necessary rule that they all should comply with? If not, which ones, for example, do and which don't ?
( Specifically: I am interested in Caputo's fractional derivation method $$D^\alpha_+ f(x)=1/\Gamma(n-\alpha)\int^x_0(x-\zeta)^{n-\alpha-1}f^{(n)}(\zeta)d\zeta$$ where $n=[\alpha]+1$.
More info. : In Caputo's method, taking the half-derivatve of an exponential function is :
$$D^{1/2}_+ \exp(kx)=k^{1/2}\exp(kx)*(1-\Gamma(1/2,kx)/\Gamma(1/2)).$$
Now, say I wish to take another derivative of the above, meaning $$D^{1/2}_+ [D^{1/2}_+ \exp(kx)]=?.$$ Can I use the $[fg]'=f'g+g'f$ rule for that? Can I simply say $$D^{1/2}_+ D^{1/2}_+f(x)=f'(x) \implies D^{1/2}_+ D^{1/2}_+\exp(kx)=k*\exp(kx) ?$$ Or am I really in truble ?)