The θ-derivative as unifying framework of a class of derivatives

Abstract

In this article, we develop a unified framework for studying some derivatives defined as limits. This framework, the θ \theta -derivative, is used to investigate the relationships between these derivatives and their relation to the ordinary derivative. It is shown that the existence of any of these derivatives is equivalent to the existence of the ordinary derivative. By using these results, we show that two derivatives that appear in the literature under different names are actually identical, and an infinite family of derivatives actually consists of only one member. We also give a unified form for the integral corresponding to these derivatives, generalize the standard analysis theorems to this setting, and relate our results to those of other researchers. Finally, we address the question of whether these derivatives should be considered fractional derivatives.