Application of Derivative in Function Limit, Series, Multiple roots, Differential and Linearization

Abstract

The derivative is one of the fundamental and practical concepts in mathematics, introduced in the early 17th century by the French mathematician Pierre Fermat. Derivative is used in solving complex calculating engineering and medical science. In this paper, derivative function is utilized in the derivatives of functions in calculations involving the exponential function  , trigonometric functions, functions limits, determining the multiple roots, differential and linearization. The significance of this research lies in demonstrating how derivatives of functions can be effectively applied in mathematics. The objective of this study is to identify specific relationships for calculating the exponential function , trigonometric functions, evaluating limits of complex and intricate functions using derivatives, determining the nature of polynomial roots as well as differential and linearization through the application of derivatives.This is an applied research and has been completed using a library research approach. Noteworthy results include established relationships for calculating functions  and  solving limits of functions using 'Hôpital' rule, and determining the nature of polynomial roots (simple, multiple, and repeated) and linearization.