A New Generalized Local Derivative of Two Parameters

We introduce a novel generalized derivative, the biparametric derivative, which constitutes an extension of the deformable derivative introduced by Ahuja Priyanka et al. (2017). This generalization is achieved when the secondary parameter, denoted by ψ, assumes the value of unity. Fundamental properties of the biparametric derivative are rigorously examined, and generalized forms of Rolle’s theorem and the mean value theorem are derived within this new framework. The biparametric integral, intrinsically associated with the biparametric derivative, is defined, and a version of the fundamental theorem of calculus adapted to this setting is established. Finally, we address and solve certain biparametric fractional differential equations as illustrative applications of the proposed operator.