The Directional Derivative in General Quantum Calculus

Abstract

In this paper, we define the β-partial derivative as well as the β-directional derivative of a multi-variable function based on the β-difference operator, Dβ, which is defined by Dβf(t)=f(β(t))−f(t)/β(t)−t, where β is a strictly increasing continuous function. Some properties are proved. Furthermore, the β-gradient vector and the β-gradient directional derivative of a multi-variable function are introduced. Finally, we deduce the Hahn-partial and the Hahn-directional derivatives associated with the Hahn difference operator.