Yuqori tartibli hosila va differensiallar

Yechish. Bu yerda hosilalar jadvali va hosila olish qoidasidan foydalanamiz:

^  ^{ x }
e^x  4x³

_ x 3 x 3
e 12x ^{2 }ln x ^

y^ 
e  4x  ln x  e  4x
ln ² x
ln x _{x . ►}
ln ² x

5. 
   
   x
   misol. ^{y  a}x ^{arctg x} funksiya hosilasi quyidagicha hisoblanadi:
   

_{ x} 
x ^ x
^a . ►

y  a  arctg x  a
arctg x  a
 ln a arctg x  _{1 x2}

Murakkab funksiyani differensiallash qoidasi bilan tanishib chiqamiz.
^{( ), ( )u bog`lixb, y f(u)u}fun^gks^xi^ya^{ x}₀ nuqtada ^{( )y} fun^f k^us^iya esa

₀ ⁽ 0 ^{)u nugqtaxdadifferensiallanuvchi bo`lsin. U holda ( ) ( ( ))y x
murakkab funksiya ham x</sup>0 <sup>nuqtada differensiallanuvchi bo`ladi:}

  
^dy _ ^dy _ ^du _{,  x  f  u} ^g^x 
dx du dx ^{0 0 0}
_{U holda} ^{dy  }^x ^{dx  f }^u ^{du . Bu yerda du  gx dx. Bu birinchi}
0 0 ⁰

differensialning invariantligi deyiladi, ya`ni murakkab funksiyada ham differensial o`z formasini saqlab qoladi.

5. 
   misol. y 3^cos5 ^{2x funksiyaning hosilasi quyidagicha hisoblanadi:}
   

y^  3^cos5 ^{2 x} ln 3cos⁵ 2x^  3^cos5 ^{2 x} ln 3 5cos⁴ 2xcos2x^ 
5ln 3  3^cos5 ^{2 x} cos⁴ 2x sin 2x2x^  10ln 3  sin 2xcos⁴ 2x  3^cos5 ^{2 x}. ^►

5. 
   misol. ^{y6 tg4 x} funksiyaning hosilasi quyidagicha hisoblanadi:
   

3 3 ^{1 24tg3 6x 24sin3 6x} _►

^{dy  4tg 6xd} ^{tg 6x} ^{ 4tg 6x}

cos² 6x

^d ^6x ^

cos² 6x

dx 

cos⁵ 6x

dx.

^{Agar yf,xa x b } funksiyaga teskari funksiya uzluksiz va differensiallanuvchi ^{x  y(u), yx  0} bo`lsa, u holda x<sup></sup><sub>y</sub> hosila ham mavjud bo`ladi:

х
х^  ^{1 .}
у у^

Masalan, hosilasi:
^{у ах} ^{а
0, аy1, 0} ^{funksiyaga teskari funksiya x loga y. Uning}

^{x } ^log
_y^ _ ¹ _ 1 _ 1 _ 1 _.








_y a
_{yx a}x
x
a^x ln a y ln a

Faraz qilamiz ^{( )y} fu^fnk^xs^iya parametrik ko`rinishda berilgan bo`lsin: ^{x  (t), y  (t), t [,  ]. Agar  ( ),  ( )x}funks^tiya^ylar dif^tferensiallanuvchi ^{va ( )  0tbo`lsa, u holda y</sup>x<sup> mavjud bo`lib quyidagicha aniqlanadi:}

t

_{y } y_t^ _ ^{ }^t  _.
^x x ^{ t }

^{Masalan, x  a cos3 t, y  b sin3 t    t  , funksiya uchun y}x^{ hosila} quyidagicha hisoblanadi:

_{y } ^yt _{ } 3bsin² t cost

  ^{b tg t, t   k , k  .}

t
^x _{x } 3a cos² t sin t a 2
^{Oshkormas ( , ) 0F xfuynksiyada y}x^{ hosilani topish uchun}

x
^{F(x, y)}^{  0  Fx (x, y)  Fy (x, y) yx  0} sifatida topib olinadi.
^{tenglamadan y}x^{ hosilani noma`lum}

Masalan, ^0arco^ts^gh^ykor^ymas^x fu^nks^iyadan
quyidagicha amalga oshiriladi:
^yx^{ hosilan topish}

^{f }^x _
1  y^{2
f } ^x ^{ 1  0  y  f }^x ^{ 1  y2}

y _cos x_^{sin x} , (cos x 0) ^ko`rinishdag^{i funksiyalarda yx hosilani hisoblash} quyidagcha amalga oshirilsdi:
^{ln y  sin x lncos x } ^{ln y  sin x lncos x}^{ 
 y  cos x ln}^{cos x} ^{ sin x sin x  .}

y
_{ } sin x ^
cos x
sin² x ^

^{y } ^{cos x} _^{cos x ln}^{cos x} ^ _{cos x} ^
 

6. 
   Yuqori tartibli hosila va differensiallar. ^{( )y}fun^fks^xi^yaning yuqori
   

^{tartibli hosilasi} y^{( )}  _ y^{( 1)n}_^{ formn ula yordamida amalga oshiriladi. Bunda har bir tartibli hosila mavjud deb faraz qilinadi. Masalan,} y ^  _{ } ^{d 2 y dy  dy } _{,
( y )} _{ }

y ^  _{ } ^{d 3 y}
_ dy _^ d ² y ^
dx²
dx _ dx _

( y )  _dx^{3
dx dx2} _ , ….
 

Bu yerda ham yuqoridagi qoidalar o`rinli. Masalan,
tartibli hosilasini hisoblaymiz:
y e^x2 funksiyaning 2-

^{y  2xex}2 ^{, y   2ex}2 ^{ 2x2xex}2  2_1  2x² _e^x2 _.

Yuqori tartibli differensiallar quyidagicha aniqlanadi:
^{d2 y d dy} - ikkinchi tartibli differensial;
^{d 3 y  d d 2 y} - uchinchi tartibli differensial;
……………………………………………
^{d n1y  dd n y} - ^n tartibli differensial.

7.