Derivative and Its Properties: Growth, Decrease Intervals, and Extreme Points

In this paper, we explore the fundamental concept of the derivative, which is pivotal in understanding the behavior offunctions in calculus. The study emphasizes the determination of growth and decrease intervals, as well as the identification of extreme points, which are crucial in optimization problems across various scientific fields, including physics, economics, and engineering. The paper discusses the mathematical principles governing derivatives, investigates the criteria for increasing and decreasing intervals, and outlines the methods to locate extreme points using the first and second derivative tests. The results obtained in this study contribute to a deeper understanding of how derivatives impact real-world scenarios, facilitating better decision-making processes in both theoretical and applied mathematics