On differentiation with respect to filters

Abstract

The article explores a generalization of the concept of the derivative of a real-valued function of one variable based on filter theory. A new construction is proposed that allows the definition of a derivative of a function with respect to a filter, which reflects the manner in which the variable approaches a given point. Unlike the classical definition, where the limit is taken via a linear approach of the argument, the new definition permits a wider range of approaches to the point, thus providing a more flexible framework for analyzing the local behavior of functions. The introduced concept includes the classical definition of the derivative as a special case when an appropriate filter is chosen. The paper presents proofs of generalized versions of basic derivative properties: linearity, product rule, quotient rule, and chain rule. In particular, it is shown that the derivative with respect to a filter satisfies the same formal differentiation rules as the classical derivative while preserving greater flexibility in how the argument approaches the point. The results obtained expand the scope of differential calculus to cases where the classical approach is either inapplicable or lacks precision or interpretative convenience. It is demonstrated that, in some situations, the derivative with respect to a filter better reflects real processes of change, such as in problems with asymmetric or constrained neighborhoods of a point. The proposed approach opens new perspectives for applications in the theory of generalized functions, measure theory, and functional analysis. The article also provides examples illustrating the application of the new concept and offers a comparative analysis with the classical theory. The presented material may be of interest to researchers in the field of mathematical analysis as well as to educators seeking to extend the traditional approach to differentiation. This work holds both theoretical and methodological value, as it introduces a new tool for further research in the field of modern limit theory.