HOSILANI HISOBLASH QOIDALARI
----------------------------

1. Ikki funksiya yigʻindisi, ayirmasi, koʻpaytmasi va nisbatining hosilasi. Aytaylik *f (x) va g (x) funksiyalari (a, b)*{\displaystyle \subset } R da

berilgan boʻlib, {\displaystyle x\_{0}}{\displaystyle \in }(a, b) nuqtada {\displaystyle f'(x\_{0})} va {\displaystyle g'(x\_{0})} hosilalarga ega boʻlsin.

Hosila taʼrifiga koʻra

{\displaystyle \lim \_{n\to \infty }}{\displaystyle {f(x)-f(x\_{0}) \over x-x\_{0}}=f'(x\_{0})}, (1)

  

{\displaystyle \lim \_{n\to \infty }} {\displaystyle {g(x)-g(x\_{0}) \over x-x\_{0}}=g'(x\_{0})} (2)

boʻladi.

1) {\displaystyle f(x)\pm g(x)} funksiya {\displaystyle x\_{0}} nuqtada hosilaga ega boʻlib,

{\displaystyle (f(x)\pm g(x))'\_{x}=f'(x\_{0})\pm g'(x\_{0})}

{\displaystyle (f(x)\pm g(x))'\_{x}=f'(x\_{0})\pm g'(x\_{0})}

boʻladi.

{\displaystyle \blacktriangleleft F(x)=f(x)\pm g(x)} deb topamiz:

{\displaystyle {F(x)-F(x\_{0}) \over x-x\_{0}}={f(x)-f(x\_{0}) \over x-x\_{0}}\pm {g(x)-g(x\_{0}) \over x-x\_{0}}.}

  

Bu tenglikda {\displaystyle x\rightarrow x\_{0}} da limitga oʻtib, yuqoridagi munosabatlarni eʼtiborga olsak. Unda

  

{\displaystyle \lim \_{n\to x\_{0}}{F(x)-F(x\_{0}) \over x-x\_{0}}=\lim \_{n\to x\_{0}}={f(x)+f(x\_{0}) \over x-x\_{0}}\pm }

  

{\displaystyle \pm \lim \_{n\to x\_{0}}{g(x)-g(x\_{0}) \over x-x\_{0}}=f'(x\_{0})\pm g'(x\_{0})}

  

boʻlishi kelib chiqadi. Demak,

  

{\displaystyle F'(x\_{0})=(f(x)\pm g(x))'\_{x}=f'(x\_{0})\pm g'(x\_{0}).\blacktriangleright }

  

2) {\displaystyle f(x)\cdot g(x)} funksiya {\displaystyle x\_{0}} nuqtada hosilaga ega boʻlib,

  

{\displaystyle (f(x)\cdot g(x))'\_{x}=f'(x\_{0})\cdot g(x\_{0})\pm f(x\_{0})\cdot g'(x\_{0})}

  

boʻladi.

  

{\displaystyle \blacktriangleleft } F{\displaystyle (x)=f(x)\cdot g(x)} deb

  

{\displaystyle {F'(x)-F(x\_{0}) \over x-x\_{0}}}

nisbatini quyidagicha

  

{\displaystyle {F(x)-F(x\_{0}) \over x-x\_{0}}={f(x)-f(x\_{0}) \over x-x\_{0}}\cdot g(x\_{0})+{g(x)-g(x\_{0}) \over x-x\_{0}}\cdot f(x)}

  

yozib olamiz. Soʻng {\displaystyle x\longrightarrow x\_{0}} da limitga oʻtib topamiz.

  

{\displaystyle \lim \_{n\to x\_{0}}{F(x)-F(x\_{0}) \over x-x\_{0}}=g(x\_{0})\cdot \lim \_{n\to x\_{0}}{f(x)-f(x\_{0}) \over x-x\_{0}}+\lim \_{n\to x\_{0}}{g(x)-g(x\_{0}) \over x-x\_{0}}\cdot f(x)=}

  

{\displaystyle =f'(x\_{0})\cdot g(x\_{0})+f(x\_{0})\cdot g'(x\_{0}).}

  

Demak,

  

{\displaystyle F'(x\_{0})=(f(x)\cdot g(x))'\_{x0}=f'(x\_{0})\cdot g(x\_{0})+f(x\_{0})\cdot g'(x\_{0})\blacktriangleright }

  

3) {\displaystyle {f(x) \over g(x)}} funksiya {\displaystyle (g(x\_{0})\neq 0)} {\displaystyle x\_{0}} nuqtada ho silaga ega boʻlib,

  

{\displaystyle {\biggl (}{f(x) \over g(x)}{\biggr )}'\_{x0}={f'(x\_{0})\cdot g(x\_{0})-f(x\_{0})\cdot g'(x\_{0}) \over g^{2}(x\_{0})}}

  

boʻladi.

  

{\displaystyle \blacktriangleleft } Modomiki, {\displaystyle g(x\_{0})\neq 0} ekan, unda {\displaystyle x\_{0}} nuqtaning biror atrofidagi {\displaystyle x} larda {\displaystyle g(x)\neq 0} boʻladi.

SHuni etiborga olib topamiz:

  

{\displaystyle {{f(x) \over g(x)}-{f(x\_{0}) \over g(x\_{0})} \over x-x\_{0}}={f(x)\cdot g(x\_{0})-f(x\_{0})\cdot g(x\_{0})+f(x\_{0})\cdot g(x\_{0})-f(x\_{0})\cdot g(x) \over g(x)\cdot g(x\_{0})\cdot (x-x\_{0})}=}

  

{\displaystyle ={1 \over g(x)\cdot g(x\_{0})}\left[{\frac {f(x)-f(x\_{0})}{x-x\_{0}}}\centerdot g(x\_{0})-f(x\_{0})\cdot {\frac {g(x)-g(x\_{0})}{x-x\_{0}}}\right].}

  

Bu tenglikda {\displaystyle x\longrightarrow x\_{0}} da limitga oʻtib, Ushbu

  

{\displaystyle \left({\frac {f(x)}{g(x)}}\right)'\_{x0}={f'(x\_{0})\cdot g(x\_{0})-f(x\_{0})\cdot g'(x\_{0}) \over g^{2}(x\_{0})}}

  

tenglikka kelamiz.{\displaystyle \blacktriangleright }

**1-natija.** Agar {\displaystyle f(x)} funksiya {\displaystyle x\_{0}} nuqtada {\displaystyle f'(x\_{0})} hosilaga ega boʻlsa, {\displaystyle c\cdot f(x)} funksiya {\displaystyle (c=const)} {\displaystyle x\_{0}} nuqtada hosilaga ega boʻlib,

  

{\displaystyle (c\cdot f(x))'\_{x0}=c\cdot f'(x\_{0})}

  

boʻladi, yaʼni oʻzgarmas sonni hosila ishorasidan tashqariga chiqarish mumkin.

**2-natija.** Agar {\displaystyle f\_{1}(x),f\_{2}(x),...f\_{n}(x)} funksiyalar {\displaystyle x\_{0}} nuqtada hosilalarga ega boʻlib, {\displaystyle c\_{1},c\_{2},...c\_{n}} oʻzgarmas sonlar boʻlsa

u holda

{\displaystyle (c\_{1}f\_{1}(x)+c\_{2}f\_{2}(x)+...+c\_{n}f\_{n}(x)'\_{x0}=c\_{1}f\_{1}'(x\_{0})+c\_{2}f\_{2}'(x\_{0})+...+c\_{n}f\_{n}'(x\_{0})}

boʻladi.

Murakkab funksiyaning hosilasi
------------------------------

**{\displaystyle 2^{0}}. Murakkab funksiyaning hosilasi.** Faraz qilaylik, {\displaystyle y=f(x)} funksiya {\displaystyle X\subset R} toʻplamda, {\displaystyle g(y)} funksiya {\displaystyle {\{f(x)|x\in R\}}} toʻplamda

berilgan boʻlib, {\displaystyle x\_{0}\in X} nuqtada {\displaystyle f'(x\_{0})} hosilaga, {\displaystyle y\_{0}\in \{f(x)|x\in X\}} nuqtada {\displaystyle (y\_{0}=f(x\_{0}))} {\displaystyle g'(y\_{0})} hosilaga ega boʻlsin. U holda 

{\displaystyle g(f(x))} murakkab funksiya {\displaystyle x\_{o}} hosilaga ega boʻlib, 

  

{\displaystyle (g(f(x)))'\_{x0}=g'(f(x\_{0}))\cdot f'(x\_{0})}

boʻladi.

{\displaystyle \blacktriangleleft g(y)} funksiyaning {\displaystyle y\_{0}} nuqtada {\displaystyle g'(y\_{0})} hosilaga ega boʻlganligidan 

  

{\displaystyle g(y)-g(y\_{0})=g'(y\_{0})\cdot (y-y\_{0})+\alpha \cdot (y-y\_{0})}

boʻlishi kelib chiqadi. Bunda 

  

{\displaystyle y=f(x),y\_{0}=f(x\_{0})} va {\displaystyle y\longrightarrow y\_{0}} da {\displaystyle \alpha \longrightarrow 0} .

Keyingi tenglikning xar ikki tomonini {\displaystyle x-x\_{0}} ga bo'lib topamiz; 

{\displaystyle {g(f(x))-g(x\_{0})) \over x-x\_{0}}=g'(f(x\_{0}))\cdot {f(x)-f(x\_{0}) \over x-x\_{0}}+a{f(x)-f(x\_{0}) \over x-x\_{0}}}

Bundan {\displaystyle x-x\_{0}} da limitga o'tib,

{\displaystyle (g(f(x)))'\_{x0}=g'(f(x\_{0}))\cdot f'(x\_{0})}

tenglikga kelamiz{\displaystyle \blacktriangleright }

HOSILALAR JADVALI
-----------------

Quyidagi sodda funksiyalarning hosilalarini ifodalovchi formulalarni keltiramiz
-------------------------------------------------------------------------------

1. {\displaystyle (C)',C=const.}
2. {\displaystyle (x\_{a})'=a\cdot x^{a-1},} {\displaystyle a\in R,x>0} {\displaystyle (x^{n})'=nx^{n-1}} {\displaystyle n\in N,x\in R}
3. {\displaystyle (a^{x})=a^{x}lna,a>0,a\neq 1,x\in R} {\displaystyle (e^{x})'=e^{x},x\in R}
4. {\displaystyle (log\_{a}x)'={1 \over xlna},} {\displaystyle a>0,a\neq 1,x\neq 0} {\displaystyle (log\_{a}\left\vert x\right\vert )'={1 \over xlna}} {\displaystyle a>0,a\neq 1,x\neq 0.} {\displaystyle (lnx)'={1 \over x}} {\displaystyle x>0.} {\displaystyle (ln\left\vert x\right\vert )'={1 \over x}} {\displaystyle x\neq 0}
5. {\displaystyle (sinx)'=cosx} {\displaystyle x\in R}
6. {\displaystyle (cosx)'=sinx,} {\displaystyle x\in R}.
7. {\displaystyle (tgx)'={1 \over cos^{2}x},} {\displaystyle x\neq {\pi \over 2}} {\displaystyle n\in R}
8. {\displaystyle (ctgx)'-{1 \over sin^{2}x}} {\displaystyle x\neq n\pi } {\displaystyle n\in R}
9. {\displaystyle (\arcsin x)'={\operatorname {1} \! \over \operatorname {\sqrt {1-x^{2}}} \!}}, {\displaystyle \left\vert x\right\vert <1}
10. {\displaystyle (\arccos x)'=-{1 \over {\sqrt {1-x^{2}}}}}, {\displaystyle \left\vert x\right\vert <1}
11. {\displaystyle (\arctan x)'={1 \over {\sqrt {1+x^{2}}}},} {\displaystyle x\in R}
12. {\displaystyle (\operatorname {arccot} )'=-{1 \over {\sqrt {1+x\_{2}}}}} {\displaystyle x\in R}
13. {\displaystyle (shx)'=chx,} {\displaystyle x\in R}
14. {\displaystyle (chx)'=shx,} {\displaystyle x\in R}
15. {\displaystyle (thx)'={1 \over ch^{2}x},} {\displaystyle x\in R}
16. {\displaystyle (cthx)'=-{1 \over sh^{2}x},} {\displaystyle x\neq 0}

MANBALAR
--------


* Differensial hisob

  

{Худойберганов\_Г\_ва\_бошқалар\_Математик\_анализдан\_маърузалар\_1 toʻplami} dan
olindi