The purpose of this paper is to establish the results of a double Dirichlet average of the generalized Bessel-Maitland function by using fractional derivative. We obtain the solution in the compact form of a double Dirichlet average of the generalized Bessel-Maitland function. Further, several special cases involving a number of well-known functions such as the Bessel-Maitland function, Mittag Leffler functions, Wright-hypergeometric functions and H-functions etc. have been established. Numerous expansions of the generalised Bessel-Maitland function, which reduces to the Mittag-Leffler function, have been studied and applied to solve a wide range of problems in physics, biology, chemistry and engineering. In the context of fractional calculus, Bessel-Maitland function, Mittag-Leffler functions, hypergeometric function and H-functions etc. have been widely studied. Since Carlson first introduced the concept of the Dirichlet average and its various forms. The Dirichlet average of elementary function like power function, exponential function etc is given by many notable mathematicians. These averages have been investigated and utilized in a number of different fields.