Do different methods of calculating fractional derivatives have to be equal? Or do they sometimes end up differently?
An example would be nice, and if possible, an explanation as too why such formulas can disagree with one another would be exceptional.
The main reason behind this is because I noted that if we could take the fractional derivative of a function through its Taylor series, this would imply that $\frac{d^q}{dx^q}e^x-\frac{d^{q+1}}{dx^{q+1}}e^x={x^{-q-1}\over\Gamma(-q)}$, which tends to $0$ as $q$ tends to become a whole number, but still, this goes against what I would expect.
And of course, explanation on the role of constants of integration would be nice.