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1THE IMPORTANCE OF USING THE DERIVATION METHOD IN STRENGTHENING THE TOPICS OF ALGEBRA
1Abdullayev A.N., 2Abdullayeva D.M.
1Finnish Pedagogical Institute of Uzbekistan, Associate Professor, Dean of the Faculty of
Applied Mathematics and Physics 2The Student of Group 307 of Mathematics and Informatics Department https://doi.org/10.5281/zenodo.13680346
Abstract. This article analyzes the importance and application of the derivation method in algebra. Consolidation of theoretical knowledge with practical examples through the derivation method helps students to understand the subject more deeply. In the analysis, the role of the method in strengthening mathematical formulas, theorems and rules, as well as its effectiveness in the educational process is considered. Theoretical concepts related to the topic and their delivery to students through the method of elicitation are highlighted, and a conclusion is made about how this method is used in the educational process, the role of the teacher in effectively conveying knowledge to students.
Keywords: definition, proof, theorem, derivation, method, derivation method, school mathematics, theoretical knowledge, reinforcement methods, mathematical analysis, educational efficiency, practical examples, mathematical formulas, theorems and rules, teaching methods.
In algebra, the deductive method is very important in learning mathematics because it allows students to draw logical conclusions and apply theoretical knowledge to practice. The derivation method can be used to teach a variety of subjects in a mathematics course, and this includes the following aspects:
The essence of reproduction. Deduction is the process of starting from general rules or principles and arriving at concrete results or conclusions. According to this method, we create new truths based on general truths (axioms, rules or theorems). For example, solving problems based on axioms and rules is common in mathematics.
Application in geometry. The subtraction method is widely used in geometry classes. Students often solve problems based on one or more theorems and rules. For example:
- Theorem and its proof: After introducing the theorems, the teacher can ask the students to prove or find a solution to the given problem using this theorem.
- Shapes and angles: Knowing that the sum of the interior angles in triangles is 180°, students can find other angles.
Application in Algebra. The derivative method is also widely used in algebra classes, for example:
- Solving equations: Students solve equations based on general rules (for example, the rule to apply the same operation to both sides of an equation).
- Mathematical Induction: Mathematical induction is also a form of induction, which is used to prove formulas or series.
The trigonometry section of mathematics, which is taught in the upper grades of the school mathematics course, is the main major section of mathematics. Due to the large number of trigonometric formulas, memorizing them can be a bit difficult for students. Therefore, if the elicitation method is used effectively during the lesson, the students will begin to understand the
formulas rather than memorizing them. In this way, they learn to derive the rest using other formulas, which in turn is useful not only for memorizing formulas, but also for logical thinking, and moving from the specific to the general.
We will take a closer look at the basic trigonometric equations, their proofs and the proofs of some formulas derived from them.
The sine and cosine of a right triangle are defined as follows:
1st definition: In a right-angled triangle, the sine of the angle a is the ratio of the leg
opposite the angle a to the hypotenuse. Thus: sina =
Definition 2. In a right triangle a as the cosine of the angle a is said to be the ratio of the
b
side attached to the angle to the hypotenuse. cosa =-.
Definition 3. In a right-angled triangle, the tangent of the angle a is the ratio of the leg opposite the angle a to the leg adjacent to the angle a. tga = ^
Definition 4: In a right-angled triangle, the catangent of the angle a is the ratio of the leg
b
adjacent to the angle a to the leg opposite the angle a . ctga = -
Using the above definitions, we prove the following theorems. Basic trigonometric equations
tga = , we find the expression of a and b from the above formulas. a = c *
cosa
sina, b = c * cosa, we make calculations by putting these expressions into tga = p In this case, it follows that tga = c sma, we have c decreasing because the numerator and denominator have
c*cosa
b
positive non-zero values. Also, since ctga = - if we put the expression of b and a in the numerator
COS&.
and denominator of the fraction, it follows that ctga = ——.
sina
Pythagorean theorem: sin2(x) + cos2(x) = 1. This theorem derives from the Pythagorean theorem and states that the square of the hypotenuse in a right triangle is equal to the square of the opposite side and the square of the adjacent side. We do the following, that is, we use the fact
sina = ^ and cosa = ^ So, we have the expression = 1. According to the
Pythagorean theorema2 + b2 = c2. If we give a common denominator to the
expression(^)2+(^)2 = 1, we get a2 + b2 = c2. The theorem has been proven.
Develop logical thinking. The method of derivation using definitions and theorems as above develops students' logical thinking skills. They learn how to apply these rules, not just memorizing the rules when solving a problem. This method helps to develop a deeper understanding and analytical approach to mathematics.
Now, let's take a closer look at the derivatives of basic elementary functions that we use a lot in mathematics and physics classes and how they are derived. Before the concept of derivative, let's talk about the concept of limit. If x^a and its values approach the number a, then let the corresponding values of f(x) approach the number A. In this case, when the number A approaches x a, the limit of the function f(x) is called and is defined as follows: lim f(x)=A. In some cases,
x^a
we say that the function f(x) tends to A when the values of x tend to .
Derivative is one of the basic concepts of mathematics, which tells how fast a function is changing. In other words, the derivative relates the change in the value of the function to the change in the independent variable (for example, (x_0)).
Definition. The derivative of the function y=f (x) is said to be the following limit (if it
f(Y± A v)_ffv}
exists). lim---. Usually, the derivative of the function y=f (x) is defined as f(x)'. The
process of finding the derivative is called differentiation. Now, using the definition of derivative, we find the derivative of the following elementary functions.
First, we find the derivative of the simple function f(x)=x. By Hm f(x+Ax) f(x)= ^^ (x+Ax) (x) = ^^ (Ax) = 1,in this case the limit of the constant number is
Ax^0 Ax Ax^0 Ax Ax^0 Ax
in any case is equal to itself, it follows that the derivative of the given function is equal to 1.
T,T . . sin(x+Ax)-sinx sinx*cosAx+cosx*sinAx-sinx . .,, .
We show that lim-= lim-Ax if we substitute zero
Ax^0 Ax Ax^0 Ax
„ .. sinAx*cosx .. sinAx
for lim-= cosx * lim-= cosx .
Ax^0 Ax Ax^0 Ax
lim sinAx = 1 is a great limit and we can take cosx Ax as a constant since it does not
Ax^0 Ax
depend on Ax. It should also be mentioned that the derivatives of the remaining elementary functions can also be derived in this way. Mathematical proofs like this are the clearest expression of the method of induction. Students learn to use theorems to prove other theorems or mathematical results. This process involves forming a logical chain based on the rules being proved.
As a conclusion, it can be said that in school mathematics, the derivation method plays an important role in the solid analysis of students' theoretical knowledge. This method provides students with a deeper understanding of mathematical concepts and their practical application. Using the deductive method, students learn to analyze what they know in a logical sequence and summarize the results. This not only strengthens mathematical knowledge, but also paves the way for their wider and deeper application. As a result, the induction method is of great importance in developing students' mathematical thinking and preparing them for the future learning process.
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