American Journal of Research in Humanities and Social Sciences ISSN (E): 2832-8019 Volume 32, | January - 2025 P a g e | 28 www.americanjournal.org THE DERIVATIVE OF SOME FUNCTIONS BY DEFINITION Abdukakhhorov Izzatillo Shavkatali ogli Uchkurgan Specialized School Teacher of Mathematics Phone: +998(93) 265 30 97 Izzatilloabduqaxxorov3070@gmail.com A B S T R A C T K E Y W O R D S By taking the derivative of some complex function by definition, it is aimed at students to learn and creatively approach the derivative of a function. Derivative, Macleron number, function, definition, property, formula. Introduction We know ( ) ( ) ( ) 2 1( ) xn x y x x   = We obtained the derivative of the function of the form from the course of elementary mathematics by logarithmizing its derivative. But we did not find the derivative according to the definition. Let's take the derivative of this function by definition. First of all, let's mention the derivative tariff. Definition: ( )f x function  ;x a b R  is a continuous function on an interval  0 ;x a b and  0( ) ;x x a b+   function increment ( )f x increment to argument x ratio 0x → If Theres is limit to the ratio ; ( )0'k f x= number ( )f x funnction 0x A number is called the point derivative of a function. We know that ( ) 0 ( ) 1 lim ln ( ) x x f x f x x→ − = that ( ) 0f x  is . appropriate. and have derivatives at that point. The following equality holds, Macroregulator: ( )f x function ( , )a b Let a function be defined on an interval, it is 0 ( , )x a b have derivatives at that point ( )`, ``, ,......... nf f f f The following equality holds, ( ) 20 0 0 0 `( ) ``( ) ( ) ( ) ( ) ... ( ) 1! 2! ! n n n f x f x f x f x f x x x x r x n = + + + + + (*) American Journal of Research in Humanities and Social Sciences Volume 32 January- 2025 P a g e | 29 www.americanjournal.org We are given a function ( ) ( ) ( ) 2 1( ) xn x y x x   = , all 0( ) 0i x  va 0( ) 0i x  in this place 1,2,3,.....,i n= for a function, we see its first-order derivative: According to the above definition, we have the following; ( ) ( ) ( ) ( ) ( ) ( ) ( )0 0 2 0 2 0 1 0 1 0 0 0 ' lim x x xn n x x x n x x x x y x x       + +  → +  + =  In this ( )0 0i x  va ( )0 0i x  1,i n= 1-condition: Let's take the derivative in the case where 2n = 2 ( ) 2 1( ) ( ) xy x x = ( ) ( ) ( ) ( )2 0 2 0 1 0 1 0 2 0 0 '( ) lim x x x x x x x y x x     +  → +  − = =  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 0 0 lim lim x x x x x x x x x x x x x x x x         +  →  → +  − +  +  − = + =   ( ) ( ) ( ) ( ) ( )2 0 2 0 2 0 1 0 1 0 0 0 1 lim lim x x x x x x x x x x S x      + −  →  → +  − = +  + =  According to the above property, we have the following equality ( ) ( ) ( ) ( )2 0 1 0 2 0 1 0' ln x x x x S    = + In this ( ) ( ) ( ) ( )2 0 2 0 1 0 1 0 0 0 lim lim x x x x x x x S z x      →  → +  − = =  From the above expression, we get the following ( ) ( ) ( ) ( )2 0 2 0 1 0 1 0 x x x x z x x    +  =  + (1) We expand the left side of equation (1) to Macler's series and lead to equation (1) and get the following expression. ( ) ( ) ( ) ( )2 0 1 1 0 2 0 1 0' ( ) x z x x x x O x     − = +  +  ( ) ( ) ( ) ( )2 0 1 1 0 2 0 1 0' x S x x x     − = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 0 2 0 1 2 0 1 0 2 0 1 0 1 0 2 0 1 0' ' ln ' x x y x x x x x x x         − = + ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 0 1 0 2 0 1 0 1 0 2 0 2 0 1 0 ' '( ) ln ' x x y x x x x x x          = +    (2) American Journal of Research in Humanities and Social Sciences Volume 32 January- 2025 P a g e | 30 www.americanjournal.org 2-Condition: Let's take the derivative of the function when 3n = ( ) ( ) ( )3 2 3 1( ) x x y x x   = According to the above definition, we have the following; ( ) ( ) ( ) ( ) ( ) ( ) ( )3 0 3 0 2 0 2 0 1 0 1 0 3 0 0 ' lim x x x x x x x x x x y x x       + +  → +  − = =  ( ) ( ) ( ) ( ) ( ) ( )3 0 3 0 2 0 2 0 1 0 1 0 0 lim x x x x x x x x x x x x       + +  → +  − +  = +  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 0 3 0 2 0 2 0 3 0 2 01 0 1 0 1 0 0 0 lim lim x x x x x x x x x x x x x x          →  → +  − + = +   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 0 3 0 2 0 2 0 3 0 3 0 1 0 2 0 0 2 0 2 0 1 lim ' x x x x x x x x xx x x y x S x x x          + + − + → +  − + +  − According to the above property (*) and Case 2, we have: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )3 0 2 0 3 0 3 0 3 0 1 0 2 0 1 0 2 0 ' ' ln ln 1 x x x x y x x x x x          = +  ( ) ( ) ( ) ( ) ( ) ( ) 2 0 1 0 1 0 3 0 2 0 2 0 ' ' ln x x x x x x        + +   (3) Thus, based on equations (2) and (3) above, we write the state of; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 0 0 0 2 0 3 0 4 0 5 0 0 1 0 2 0 3 0 4 0' x x x xn n n n x x x x ny x x x x x               =    ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 1 0 0 0 2 0 1 0 1 0 1 0 ' ln ln 1 xn n nx x n n n n x x x x x        − − − −     +     ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 0 2 0 0 1 0 2 0 1 0 3 0 1 0 2 0 ' ' ln ln ln ln n n n n n n n x x x x x x x x x          − − − − − −  + +  ( ) ( ) ( ) ( )3 0 1 0 4 0 3 0 ' ln ln n n n x x x x     − − −  + +  American Journal of Research in Humanities and Social Sciences Volume 32 January- 2025 P a g e | 31 www.americanjournal.org ( ) ( ) ( ) ( )4 0 1 0 5 0 4 0 ' ln ln n n n x x x x     − − −   + +    ( ) ( ) ( ) ( ) ( ) 2 0 1 0 1 0 2 0 1 0 ' ' ln x x x x x        + +    The above equality can be proved by the method of mathematical induction. Example: of the function in the form of 2 (2 1) 3( ) ( ) xxy x x x += + Let's take the first order derivative from the formula given above 0 1x = Exactly ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )3 0 2 0 3 0 3 0 3 0 1 0 2 0 1 0 2 0 ' ' ln ln 1 x x x x y x x x x x          = +  ( ) ( ) ( ) ( ) ( ) ( ) 2 0 1 0 1 0 3 0 2 0 2 0 ' ' ln x x x x x x        + +   13 1 3 2 3 2 3 '(1) 2 3 ln(2) ln(3) ln(2) 24 ln(2) ln(3) ln(2) 3 2 3 2 y     =   + + =  + +        References 1. Matematikanaliz I-qismT.Azlarov H. Mansurov. 2. MatematikanalizII-qismT.Azlarov H. Mansurov. 3. ziyonet.uz.