Derivative generals and some applications

In this thesis, the concept of differentiation is examined in the context of classical derivatives, Fréchet derivatives, and quantum derivatives, with the aim of introducing a new derivative definition as a generalization of these types of derivatives. The study specifically aims to address the limitations of the classical notion of limits in non-linear and non-limit-based operators. In this framework, the usage of classical derivatives in linear operators, Fréchet derivatives in non-linear operators, and quantum derivatives in non-limit-based operators are analyzed, and the shortcomings of existing methods are identified. To overcome these limitations, a new derivative definition termed as "quantum Fréchet derivative" is developed by applying the quantum derivative to the Fréchet derivative. The mathematical structure of this new derivative is thoroughly explored in the dissertation, and its theoretical validity is supported by various propositions and theorems. It is demonstrated that the new derivative satisfies fundamental properties, such as assigning a zero derivative to constant functions, preserving the differentiability characteristics of certain composite functions, and producing consistent results when applied to polynomial functions. Furthermore, the application of this new derivative in analytical tools such as Taylor series is discussed and illustrated through examples. In conclusion, this dissertation provides a generalization that addresses gaps in the existing literature on derivatives, offering a novel perspective on the use of derivatives in engineering, physics, and other applied fields. The study also opens new avenues for future research, particularly in exploring the relationship between the newly defined derivative and the concept of integration, as well as further advanced analyses.