مروری بر تحلیل نظری مشتقات توابع متعالی در ریاضیات عالی

Abstract

Abstract
Transcendental functions, as one of the fundamental components of mathematical analysis, play a crucial role in the development of advanced concepts in higher mathematics. The derivatives of these functions, especially in examining analytical behavior, continuity, smoothness, and higher-order differentiability, are of particular significance. The main objective of this research is to analyze the theory of derivatives of transcendental functions and explain their position within the conceptual framework of advanced mathematics. In this study, the basic definitions of transcendental functions, including exponential, logarithmic, trigonometric, and certain special functions, are first reviewed, and then the rules and structures of differentiation for these functions are analyzed theoretically. The research methodology is fundamental, with an analytical-descriptive approach, and the data are collected through library studies and an examination of credible domestic and international scientific sources. The primary focus of the research is on analyzing the analytical relationships, theorems, differentiation formulas, and the behavior of higher-order derivatives of transcendental functions. In this context, concepts such as uniformity, convexity, convergence of power series related to derivatives, and their connection to the structure of mathematical analysis are emphasized. The findings of the research indicate that the derivatives of transcendental functions, in addition to their fundamental role in mathematical analysis, provide an effective tool for a deeper understanding of the structure of continuous and analytic functions. Moreover, the theoretical analysis of these derivatives provides a suitable basis for the development of more advanced discussions in higher mathematics, including real and complex analysis. The main innovation of this study lies in presenting an analytical framework to explain the position of the derivatives of transcendental functions relative to previous sources, which can help clarify the contribution of this study to the scientific literature. Additionally, the results of the research can serve not only as a theoretical framework for future studies in mathematical analysis but also as a significant tool in teaching advanced mathematical concepts and applications related to the analysis of function behaviors.
Keywords: transcendental functions, exponential functions, logarithmic functions, trigonometric functions, derivatives