A SPHERE AND ITS FRAGMENTS

Аннотация

This article provides a detailed explanation of the geometric and physical meaning of the concept of derivative. From a geometric point of view, the derivative is interpreted as a gradient that determines the direction of the tangent line at each point of the curve. In physics, the derivative is often used to express quantities such as motion, velocity, and acceleration. The article reveals the importance of the derivative through real-life examples along with theoretical concepts. It also discusses the practical application of the derivative and what methods can be effective in explaining it to students in education. Keywords: Derivative, tangent, gradient, velocity, acceleration, physical meaning, geometric interpretation, differential, curve, analysis. Today, the educational process is required to be closely related to the development of science and technology. However, this does not mean that students should be taught all the innovations created by mankind. Each educational institution should select the materials it needs from the achievements of science and technology. One of such important topics is information about the integral and integral of a function. The geometric and physical content of these concepts is covered in many literatures, and they are effectively used in the educational process. As a result of the development of mathematics, the concepts of the integral and derivative of a function have also appeared, and scientific research is being conducted on their properties. Today, it is becoming urgent to introduce these concepts into the educational system. In particular, it is advisable to teach these topics through special courses or additional lessons in academic lyceums and higher educational institutions. This can develop students' logical thinking skills and increase their interest in science. During its development, humanity, while providing knowledge to the younger generation, tries to teach the basics of sciences, taking into account the requirements of its own activities and development. Therefore, the task of providing students with in-depth knowledge of mathematics, along with all other knowledge, and its scientific implementation are among the main issues. 143 SCIENCE AND INNOVATION IN THE EDUCATION SYSTEM International scientific-online conference In this regard, the methodology of teaching mathematics occupies one of the main places. The word "methodology" is derived from the Greek word "method" or "method". The science of mathematics teaching methodology (method) is called the branch of pedagogy that studies and researches the methods and laws of teaching mathematics at a certain level of its development in accordance with the educational goals set by society. Mathematical methodology is one of the main branches of pedagogy and didactics, and is an independent science that studies the laws of teaching and learning mathematics that correspond to the goals of education at the level of development of our society. The concept of mathematical methodology was first described in the work of the Swedish pedagogue and mathematician G. Pestalozzi, written in 1803, “The Demonstrative Study of Numbers.” Starting from the first half of the 17th century, issues related to the methodology of teaching mathematics were addressed by Russian scientists, such as Academician S.E. Guriev (1760-1813), and from the first and second half of the 18th century, by N.I. Lobachev (1792- 1856), I.N. Ulyanov (1831-1886), L.N. Tolstoy (1828-1910), and the outstanding methodologist-mathematician S.I. Shokhor-Trotsky (1853-1923), A.N. Ostrogrotsky, and others. They looked at mathematics from a scientific point of view and developed its progressive foundations. For example, A.N. Ostrogradsky wrote: “Consciousness arises after observation, consciousness "based on the real, existing world." In addition, in covering the topic, students are shown practical application aspects by explaining its geometric and physical content. This helps students understand the importance of science in life. The derivative is one of the basic concepts in mathematics, which is widely used in mathematics, physics and other sciences to analyze various problems, in particular, the speed of various processes. In mathematical analysis, the derivative is one of the basic concepts that determines the rate of change of a variable quantity. Understanding the theoretical foundations of the derivative allows us to apply it to various practical problems. In this article, we will consider the essence of the derivative from two main perspectives - geometric and physical. The derivative is the rate of change of a function at a certain point. Geometrically, it is the slope of the tangent line drawn to the curve corresponding to the function at this point. By determining how the derivative changes, you can analyze how the graph will look. 144 SCIENCE AND INNOVATION IN THE EDUCATION SYSTEM International scientific-online conference Differential calculu