A New Approach to Generalized Fractional Derivatives

The author (Appl. Math. Comput. 218(3):860-865, 2011) introduced a new fractional integral operator given by, \[ ({}^ρI^αₐ₊f)(x) = {ρ1- α}{Γ(α)} ∫^x_a {τρ-1 f(τ) }{(x^ρ- τ^ρ)1-α}\, dτ, \] which generalizes the well-known Riemann-Liouville and the Hadamard fractional integrals. In this paper we present a new fractional derivative which generalizes the familiar Riemann-Liouville and the Hadamard fractional derivatives to a single form. We also obtain two representations of the generalized derivative in question. An example is given to illustrate the results.