Here, the Riemann–Liouville fractional derivative operator concept and a new and interesting extended form of Appell’s functions are discussed. We discovered new formulae for fractional derivatives of various well-known functions in terms of new extended Appell’s hypergeometric functions of two variables and Lauricella hypergeometric functions of three variables, with a view toward the analytic properties and application of the new Riemann–Liouville-type fractional derivative operator. Additionally, we defined the Mellin transformations of that function. Next, we created generating functions for generalized extended hypergeometric functions to validate our new operator using an extended Riemann–Liouville fractional derivative operator and a new definition of extended Appell’s functions.