Abstract
This article discusses the properties of complex functions, methods This article discusses the properties of complex functions, ways to use the physical and geometric meaning of the derivative when solving problems, ways to find an effective solution using the derivative when solving applied, including socio-economic problems. The study included an analysis of how students solve problems based on a pedagogical experiment, an analysis of documents to describe the mathematical methods chosen when solving problems using the interview method, as well as algorithms used in solving them. As a result of the study, special attention was paid to methodological recommendations for studying the elements of mathematical analysis in a school mathematics course, as well as methods for solving typical mathematical problems encountered in state exams and olympiads in mathematics in grades 9-11. The results of scientific research on analyzing and demonstrating solutions to non-standard and Olympic problems of various levels of complexity together with the student are presented. As a result of the study, several methods were proposed as an effective way to solve non-standard problems associated with elements of mathematical analysis. The result of scientific research can be used as auxiliary material when solving non-standard problems at school, lyceum or gymnasium, where mathematics is taught in depth.