Yuqori tartibli hosila va differensiallar

Funksiya differensiali. Agar ^y

aniqlangan bo`lib, uning ^yorttirmasini
^{f (x) x0 nuqtaning}  ^atrofida

      ( ^{)y A x x x}
ko`rinishda tasvirlash mumkin bo`lsa, u holda ^{( )y} fu^fnk^xs^iya differensiallanuvchi A xesa uning differensiali deb ataladi. Bu yerda
^x0 ^{nuqtada
( 0 )A x}xga

bog`liq emas, ^{lim  (}
 0
)  0x^.

Funksiya differensiali quyidagicha yoziladi: ^{  ,  ( )d}. ^{y df Adx A}

5. 
   misol. ^{y x2} funksiya differensiallanuvchi. Haqiqattan ham
   

y  ^x  x^2  x²  2x  ^x^ ^x^2  2x ^x^ o^x^. ►

	3-TEOREMA
^y^^f^(^x^)funksiya ^xnuqtada differensiallanuvchi bo`lishi uchun u 0 bu nuqtada hosilaga ega bo`lishi zarur va yetarli va quyidagicha bog`langan ^dy^(^^)^f^^x^x. 0 Agar funksiya^a^(^b^, )intervallning har bir nuqtasida differensiallanuvchi bo`lsa, u holda bu funksiya^a^(^b^, )intervalda differensiallanuvchi bo`ladi.

^{      ( )y}for^Amu^xlada ^{x  (x )x}qo`s<sup>x</sup>hiluvchi cheksiz kichik miqdor bo`lgan uchun bu formulani quyidagicha yozish mumkin:
y  f ^(x₀ )Ax  f (x₀  x)  f (x₀ )  f ^(x₀ )x .
Bu formuladan taqribiy hisoblarda foydalanish mumkin.

5. 
   misol. y ^funksiyaning 90ⁿx^{uqtadagi qiymatini toping.}
   

Yechish. Bu yferxda (x) , ₀ ^x81, x9^{deb faraz qilamiz. U holda}

^{( )fx4 81  3 ,} f ^(x)  ^{, f (x ) 
 3  1} ,

 3,083. ►

₁ 1
0 0 4  3³ 12

4. 
   ^{Hosilaning geometrik ma`nosi. M x  f ;x  } nuqtada ^{y f x}
   

0 0

funksiyaga o`tkazilgan urinma deb MN kesuvchining (2-rasm) N nuqtasi M nuqtaga funksiya grafigi bo`ylab ixtiyoriy ravishda yaqinlashishini qabul qilamiz. Bunda

0. dx

2-rasm
^{f x  qiymatM x  f ;x  } nuqtada ^{f x} funksiyaga o`tkazilgan urinmaning

0 0 0
^{  (} 0 ^{)tgburchf akxkoeffitsientini bildiradi (2-rasm).
M x  f ;x   nuqtada f x} funksiyaga o`tkazilgan urinma tenglamasi quyidagi
0 0

ko`rinishga ega bo`ladi:

^{y  f} ^x0  ^{ f }^x0 ^{x  x0}  ^{urinma tenglamasi}^.

5. 
   misol. ₀ ^(4;2)M nuqtada ( )f x
   

tenglamasini tuzing.
fxunksiyagao`tkazilgan urinma

^Yechish. f x    2, f ^x  ¹ , f ^x  ^{1  1 .}
0 2 0 2 4

Demak urinm^ya tenglam^a^xsi:^{ 2} 1 ^
4
^4 (3-rasm). ►

5. 
   misol. ^0,O0 nuqtada y funksiyaga o`tkazilgan urinma tenglamasi
   

^{0f } bo`lgani uchun u vertikal to`g`ri chiziq bo`ladi (4-rasm). ►

3-rasm 4-rasm

5. 
   Hosila olish va differensiallash qoidalari. ^{( )f} v^xa ^{( )g}fu^xnksiyalar ^x
   

^{nuqtada differensiallanuvchi bo`lib,</sup> k const<sup>bo`lsin. U holda:}
1. _{ f x gx}^ _{ f} _x^_{ g}_x;^{ df x gx df x dgx}. ^2. kf x^  kf ^x^{; dkf x k df x.}

^3.  f xgx^ 
.
_{f xgx}_{f xgx}; df xgx df xgx f xdgx

_

2  

2

4. _ ^{f x}_ ^ f ^xgx f xg^x ;  _{f x} ^{df xgx  f xdgx
  0g}^{. x}

gx
g x
_d gx _
g x _

   

Funksiyaning hosilasi va differensialini hisoblashda zarur bo`ladigan elementar funksiyalarning hosilalari jadvalini keltiramiz:
^{ C1)  C} 0,const .

2) ^{x   x1,  Rx1, x nx0, nn N}


x^n1,R  ,  ¹

1) ^{ax    x ln a,   0, a  1, xRe}1^{, ex  x} x ^{,R  .}1

2) log x^  ^{1 ,}
a  0, a  1, x  0.

^a x ln a

log x^  ^{1 ,}


a  0, a  1, x  0.

a
 lnx^
 lnx ^
x ln a
¹ , x 0.
x
¹ , x 0.
x

 ⁵⁾ sin x^{ cos x, x R1.}

 ⁶⁾ cos x^{ sin x, x R1.}

7) tg x^  ¹ ,
cos² x
^{x  }  n, n  Z.
2

8) ctg x^   ^{1 ,}
sin ² x
x  n, n  Z.

9. 
   ^{arcsin x}^{  1 ,}
   

x  1.

9. 
   arccosx^   ¹ ,
   
10. 
    arctgx^  ^{1 , x}
    

1 x²
x  1.


R¹.

9. 
   arcctgx^   ^{1 , x}
   

1 x²


R¹.

 ¹³⁾ shx^
 ¹⁴⁾ chx^
 15) thx^


ch x, x


sh x, x
¹ , x ch² x


R¹.


R¹.


R¹.

16) cth x^   ^{1 ,}
sh² x
e^x  4x³


x  0.

5. misol.

y  ^{funksiyaning hosilasini hisoblang.}
ln x