Volume 15 Issue 05, May 2025
Impact factor: 2019: 4.679 2020: 5.015 2021: 5.436, 2022: 5.242, 2023:
6.995, 2024 7.75
http://www.internationaljournal.co.in/index.php/jasass
174
APPLY THE RESULT TO ECONOMIC PROCESSES.
Iqbaljon Khaydarov
Kokand University, Teacher, Department of Digital
Technologies and Mathematics.
Abstract:
This article discusses the importance of the concept of derivative in economic
processes, its role in optimization and marginal analysis. It explains how the concepts of
marginal revenue, profit and cost are determined through derivatives, as well as methods for
calculating price elasticity. The practical application of the derivative in economic processes is
examined through examples, and the importance of the derivative in the strategies for
maximizing the profit of companies is emphasized.
Keywords:
Derivative, economic processes, optimization, limit analysis, elasticity.
Аннотация:
В данной статье рассматривается значение понятия производной в
экономических процессах, её роль в оптимизации и предельном анализе. Объясняется, как
с помощью производных определяются предельный доход, прибыль и издержки, а также
методы расчёта ценовой эластичности. Практическое применение производной в
экономике проиллюстрировано на примерах, подчёркивается важность производной в
стратегиях максимизации прибыли компаний.
Ключевые слова:
производная, экономические процессы, оптимизация, предельный
анализ, эластичность.
Introduction
Mathematics and economics are closely related, and one of the main tools of mathematical
analysis - the concept of the derivative - is widely used in modeling and analyzing economic
processes. The use of the derivative is of great importance in improving the efficiency of
economic processes, making optimal decisions, and forecasting processes. The derivative is used
to determine the speed and direction of economic changes and plays an important role in
business and macroeconomic policy.
Concepts developed on the basis of the derivative in economic models and theories, such as
marginal revenue, marginal cost, and marginal profit, are key indicators in making business
decisions. Marginal analysis is widely used by enterprises to optimize production volumes,
effectively use resources, and set prices. In addition, the derivative is also used to define the
concept of elasticity, which allows us to determine the sensitivity of consumers to demand.
Therefore, this article examines the role of derivatives in economics, their practical applications,
and methods of analysis. Based on scientific sources, theoretical concepts, and practical
examples, it examines how derivatives help in understanding and managing economic processes.
The results of the study provide a deeper understanding of the effectiveness of derivatives in
economic modeling and strategic decision-making.
Methodology
The article presents theoretical and practical analyses of the application of derivatives in
economic processes based on mathematical analysis and economic modeling methods. The
Volume 15 Issue 05, May 2025
Impact factor: 2019: 4.679 2020: 5.015 2021: 5.436, 2022: 5.242, 2023:
6.995, 2024 7.75
http://www.internationaljournal.co.in/index.php/jasass
175
research shows how basic economic concepts such as marginal analysis, optimization, and
elasticity can be expressed using derivatives through examples.
The theoretical foundations of economic modeling were developed using the principles of
economic analysis put forward by Samuelson and Nordhaus (2009). Also, the mathematical
expressions of the concepts of marginal profit and cost were studied based on the methods of
microeconomic analysis proposed by Varian (1992). At the first stage, the use of derivatives in
economic models was analyzed. How indicators such as marginal revenue and marginal cost are
calculated based on derivatives and their role in the development of economic strategies was
studied based on examples. For example, a company can optimize profits by increasing or
decreasing the volume of product production.
In the second stage, elasticity analysis was conducted using the derivative. Demand elasticity is
used to determine the relationship between the price of a product and the volume of consumer
demand. In this stage, demand elasticity was calculated using the derivative at different price
levels and how consumers responded to price changes was studied. Silberberg and Suen (2001)
emphasized the use of the derivative in economic elasticity analysis and emphasized its
importance in strategic decision-making. In the third stage, optimization problems were solved.
In the process of economic modeling, optimal points were identified using the second-order
derivative analysis. In particular, methods for maximizing profit and minimizing costs were
studied. Differential equations and boundary conditions were used in these processes. Production
and pricing problems were analyzed based on research by Chiang (1984) on the importance of
the derivative in economic modeling.
The results were compared with economic models and real-life data, and the role of the
derivative in optimizing economic processes was determined. The research methods used were
mathematical differentiation, limit analysis, and statistical analysis based on empirical
observations.
Results
The results of this study clearly confirm the role and importance of the derivative in the
application of mathematical analysis to economic processes. Based on the formulas of marginal
profit, marginal revenue and marginal cost studied during the study, the possibilities of
companies to maximize their profits were identified. Static and experimental analyses have
shown that the determination of the growth rate of economic variables through the derivative and
the determination of the point of infinity in the optimization process are important. At the same
time, the application of the concept of the derivative in economic models and theories serves to
achieve reliable results in the calculation of marginal analysis and price elasticity.
The application of the MR = MC condition based on boundary analysis helped companies
determine the optimal production volume. This approach made it possible to reduce production
costs, increase profit levels, and use resources efficiently. Using the product of price elasticity,
consumer demand sensitivity was determined and strategies were developed to increase market
competitiveness.
The results of the study were noteworthy, as scientists Abdurakhmonov (2010), Shermatov
(2012), and Millatov (2015). As they noted, the use of mathematical analysis tools accurately
reflects the complexity of economic processes and allows for high efficiency in making business
decisions. The integration of Yunusov's (2020) approaches and traditional theories opened up
new horizons in the formation of economic strategies.
Volume 15 Issue 05, May 2025
Impact factor: 2019: 4.679 2020: 5.015 2021: 5.436, 2022: 5.242, 2023:
6.995, 2024 7.75
http://www.internationaljournal.co.in/index.php/jasass
176
In addition, the study found that the use of derivatives is important in the formation of not only
theoretical foundations, but also practical strategies in the process of making economic decisions.
In this regard, it was observed that the ideas put forward by economists, in particular, the views
of Karimov (2018) and Yunusov (2020), are consistent with practical results. As they noted,
using the derivative methodology, it is possible to increase production efficiency, quickly
respond to market changes, and optimally allocate resources. In addition, this approach has its
place in ensuring the stability of the economic system. This approach helps not only to
strengthen theoretical foundations, but also to optimize the process of making practical decisions.
All this serves to significantly accelerate economic development.
The use of derivatives in economic processes helps to solve important problems such as
maximizing or minimizing profit. The following is a discussion of the issue of maximizing and
minimizing profit to illustrate how derivatives are used in real economic situations.
Problem:
The total profit of a company is represented by the following function:
P(x)=−2x
2
+40x−100
where x is the number of units produced (thousands of units). We use derivative analysis to
determine the optimal points for the company to maximize or minimize its profit.
Solution:
1. Find the derivative to maximize and minimize the profit.
To determine the maximum and minimum points, the first derivative of the function is taken:
P′(x)=
�
��
(−2x
2
+40x−100)=−4x+40
To determine the extremum points, the derivative is set to zero:
−4x+40=0
x=10
This value can be the point at which the company's profit is maximized or minimized.
2. Determining the type of extremum using the second derivative
The second derivative of a function is:
P′′(x)=
�
��
(−4x+40)=−4
Since the second derivative is negative:
P′′(10)=−4<0
This result shows that the function reaches a maximum at x=10.
Result:
• The company achieves maximum profit by producing 10,000 units of the product.
• Minimum profit is achieved by increasing or decreasing production indefinitely.
• Derivative analysis helps companies determine the optimal production volume and develop
economic strategies.
Conclusion:
The study proved that the derivative is of great importance in the analysis and optimization of
economic processes. Marginal profit, income and cost indicators, as well as price elasticity are
important elements in the economic model. These approaches have been widely used in the
formulation of business decisions, improving production efficiency, and determining the
sensitivity of consumer demand. The results create a solid foundation for further comprehensive
research and the development of economic theories. According to scientists, this methodological
Volume 15 Issue 05, May 2025
Impact factor: 2019: 4.679 2020: 5.015 2021: 5.436, 2022: 5.242, 2023:
6.995, 2024 7.75
http://www.internationaljournal.co.in/index.php/jasass
177
approach plays an important role in solving modern problems of the economy and making
strategic decisions, as well as in achieving practical results.
List of used literature.
1. Samuelson, P. A., & Nordhaus, W. D. (2009).
Economics
(19- chi nashr). McGraw-Hill
Education.
2. Varian, H. R. (1992).
Microeconomic Analysis
(3- chi nashr). W. W. Norton & Company.
3. Silberberg, E., & Suen, W. (2001).
The Structure of Economics: A Mathematical Analysis
.
McGraw-Hill.
4. Chiang, A. C. (1984).
Fundamental Methods of Mathematical Economics
. McGraw-Hill.
5.
Abdurahmonov, X. (2010).
Iqtisodiy modellashtirishda matematik analiz
. Toshkent:
O‘zbekiston Noshirligi.
6. Shermatov, R. (2012).
Chegaraviy tahlil va iqtisodiy qarorlar
. Toshkent: O‘zbekiston
Noshirligi.
7. Millatov, M. (2015).
Biznes strategiyalarida hosilaning qo‘llanilishi
. Toshkent: O‘zbekiston
Noshirligi.
8. Karimov, N. (2018).
Iqtisodiy tizimda innovatsion yondashuvlar
. Toshkent: O‘zbekiston
Iqtisodiyot Noshirligi.
9. Yunusov, Z. (2020).
Bozor o‘zgarishlarining matematik tahlili
. Toshkent: O‘zbekiston
Iqtisodiyot Instituti Noshirligi.
10.
Haydarova K. ROBOTOTEXNIKADA SENSORLAR VA AKTUATORLAR. MA’LUMOT
CHIQARUVCHI DISPLAY TURLARI //QO ‘QON UNIVERSITETI XABARNOMASI. –
2024. – Т. 13. – С. 366-371.
11. Haydarova K. ROBOTOTEXNIKA: IT SOHASIDAGI AHAMIYATI VA O’RGANILISH
DARAJASI //University Research Base. – 2024. – С. 1004-1006.
12.
Azimova, T. E. (2024). OLIY TA’LIMDA ELEKTRON TA’LIM RESURSLARIDAN
FOYDALANISHNING AHAMIYATI.
Kokand University Research Base
, 406-408.
13.
Nuritdinov, J. T., & Azimova, T. E. (2024). AYRIM SONLARNI KO
‘PAYTIRISHNING SODDA USULLARI.
Kokand University Research Base
, 423-428.
14.
FA, Nuraliev, and Kuziev Sh S. "THE COEFFICIENTS OF AN OPTIMAL
QUADRATURE FORMULA IN THE SPACE OF DIFFERENTIABLE FUNCTIONS."
Uzbek
Mathematical Journal
67.2 (2023).
15.
Nuraliev F. A., Kuziev S. S. Optimal Quadrature Formulas with Derivative in the Space:
Optimal Quadrature Formulas with Derivative in the Space //MODERN PROBLEMS AND
PROSPECTS OF APPLIED MATHEMATICS. – 2024. – Т. 1. – №. 01.
16.
Qo’Ziyev S. S., Tillaboyev B. S. O. TALABALARDA IJODKORLIKNI
RIVOJLANTIRISHDA AXBOROT KOMMUNIKATSION TEXNOLOGIYALARNING O
‘RNI //Oriental renaissance: Innovative, educational, natural and social sciences. – 2021. – Т. 1.
– №. 10. – С. 344-352.
17.
Shadimetov K., Nuraliev F., Kuziev S. Coefficients and errors of the optimal quadrature
formula of the Hermite type //AIP Conference Proceedings. – AIP Publishing, 2024. – Т. 3147. –
№. 1.
18.
Shadimetov K., Nuraliev F., Kuziev S. Optimal Quadrature Formula of Hermite Type in
the Space of Differentiable Functions //International Journal of Analysis and Applications. –
Volume 15 Issue 05, May 2025
Impact factor: 2019: 4.679 2020: 5.015 2021: 5.436, 2022: 5.242, 2023:
6.995, 2024 7.75
http://www.internationaljournal.co.in/index.php/jasass
178
2024. – Т. 22. – С. 25-25.
