DERIVATIVE OF A FUNCTION AND ITS APPLICATIONS IN ECONOMICS

Аннотация

В данной статье рассматривается значение понятия производной функции при решении экономических задач. В частности, анализируется применение производной при построении уравнения касательной к графику функции, расчёте скорости и ускорения, затрат на производство, эластичности функции, эластичности спроса по доходу и цене. В статье подчёркивается необходимость методического обогащения преподавания математики в экономических направлениях высшего образования. КЛЮЧЕВЫЕ СЛОВА:Производная, экономические задачи, эластичность функции, эластичностьспроса,цена,доход,математический анализ,методикапреподавания ANNOTATION. This article highlights the significance of the derivative concept in solving economic problems. Specifically, it analyzes the use of derivatives in constructing tangent line equations, calculating speed and acceleration, production costs, function elasticity, and theelasticity of demand with respect toincome and price. The article emphasizes theneed for methodological enrichment of teaching mathematics in economics-oriented higher education programs. KEYWORDS. derivative, economic problems, function elasticity, demand elasticity, price, income, mathematical analysis, teaching methodology INTRODUCTION. The introduction of new standards of higher education in the Republic of Uzbekistan is a necessity dictated by the current situation and the rapid development of science and technology, as well as the integrated development of all sectors of the national economy. It is the new standards that set the goal of forming a personality witha set ofqualities that allowaperson to besuccessful inthe21st century. The purpose of higher education is to create conditions for student self-realization in the educational process, to develop the ability to pose practical problems in one’s specialty, the ability to choose methods for solving a given problem, to solve, analyze and apply solutions to practical problems, using the latest achievements of science and technology. To achieve these goals, we consider it advisable to use professionally oriented modern page1605 INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE ISSN:2692-5206,ImpactFactor:12,23 AmericanAcademicpublishers,volume05,issue04,2025 Journal: https://www.academicpublishers.org/journals/index.php/ijai pedagogical technologies in teaching all subjects at universities, taking into account the specifics of thesubject and specialty. The subject of mathematics has its own special place in this process. Here you should definitely take into account how much attention the government and state of Uzbekistan pays to the issue of education in general, as well as the teaching of mathematics at all levels of education and theintegration ofscience and production. The emergence of students' interest in mathematics depends to a large extent on the methodology of its teaching, on the chosen style of communication with students, and on the extent to which theapplied problems ofthis particular specialty are demonstrated. Whensolving many problems in science, technology, and economics, the concept of a derivative function is encountered. For example, when drawing up an equation for a tangent to a curve, when calculating speed and acceleration, when studying and plotting functions, when calculating production costs, elasticity of a function, elasticity of demand relative to income, relative to price, and others. Therefore, studying the concept of derivative is one of themostimportant issues in mathematics. METHODS. Definition 1. Let be a function defined outside some neighborhood ofthepoint .Thederivative of thefunction at thepoint is thenumber equal to f x (1) x0 f x x0 f x −f(x0) If this limit existsand isdenoted by . xl→imx0 x−x0 To find thederivativeof agiven function, you mustperform the followingwork: 1) Let’s assign an increment tof'(txh 0 e)free variable x, then the function y takes on an increment ,i.e. ). ∆x 2) yLe+t’s∆yfin=dft(hxe+inc∆rxement ofthefunction : .∆y 3) Find theratio of theincrement of thefunction to theincrement of theargument: ∆y = f x+∆x −y = f x+∆x −f(x) . ∆y f x+∆x −f(x) 4) ∆Lxe=t us cal∆cxulate the limit of the ratio of the increment of the function to the increment of the argument as ,i.e. ∆x → 0 . ∆y f x+∆x −f(x) ∆lxim→0∆x = ∆lxim→0 ∆x If this limit exists, then it is called the derivative of the function at the point andis denoted by or ,i.e. f(x) x0 (2) y' f'(x0) ∆y If limit f (2 x+ ) ∆ e x x − is f t ( s x , ) then' as . This means that or ∆lxim→0∆x = ∆lxim→0 ∆x = f x0 = y' . Consequently, the derivative at a point exists only for functions that are continuous at thepoint (and even then not∆fyor→al0l). ∆x → 0 f x −f(x0) → 0 f x → f(x0) x0 page1606 INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE ISSN:2692-5206,ImpactFactor:12,23 AmericanAcademicpublishers,volume05,issue04,2025 Journal: https://www.academicpublishers.org/journals/index.php/ijai Limit (2) may not exist at all or may exist for some values of x and not exist for other values. At those