Note on the generalized conformable derivative

. We introduce a definition of a generalized conformable derivative of order α > 0 (where this parameter does not need to be integer), with which we overcome some deficiencies of known local derivatives, conformable or not. This definition allows us to compute fractional derivatives of functions defined on any open set on the real line (and not just on the positive half-line). Moreover, we extend some classical results to the context of fractional derivatives. Also, we obtain results for the case α > 1.