A New Approach to Generalized Fractional Derivatives

The author \mbox{(Appl. Math. Comput. 218(3):860-865, 2011)} introduced a new fractional integral operator given by, $$\left({}^{\rho }{\mathcal{I}}_{a+}^{\alpha }f\right)(x)=\frac{{\rho }^{1-\alpha }}{\mathrm{\Gamma }(\alpha )}{\int }_a^x\frac{{\tau }^{\rho -1}f(\tau )}{(x^{\rho }-{\tau }^{\rho }{)}^{1-\alpha }}d\tau ,$$ 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.