Hosilaning iqtisodga tadbiqlari

Hosilaning ta’rifi, geometrik va mexanik ma’nolari
Hosilaning ta’riflari
EMBED Equation.3 funksiya EMBED Equation.3 intervalda aniqlangan bo‘lsin. Ixtiyoriy EMBED Equation.3 nuqtani olamiz va bu nuqtada EMBED Equation.3 argumentga EMBED Equation.3 orttirma ( EMBED Equation.3 ) beramiz. Bunda funksiya EMBED Equation.3 orttirma oladi.
1-ta’rif. Agar EMBED Equation.3 limit mavjud va chekli bo‘lsa, bu limitga EMBED Equation.3 funksiyaning EMBED Equation.3 nuqtadagi hosilasi deyiladi EMBED Equation.3 (yoki EMBED Equation.3 yoki EMBED Equation.3 ) kabi belgilanadi.
Shunday qilib,
EMBED Equation.3 . (6)
Agar EMBED Equation.3 ning biror qiymatida EMBED Equation.3 EMBED Equation.3 bo‘lsa, u holda funksiya EMBED Equation.3 nuqtada musbat ishorali (manfiy ishorali) cheksiz hosilaga ega deyiladi. Shu sababli 1-ta’rif bilan aniqlanadigan hosila chekli hosila deb yuritiladi.
Misollar. 1. EMBED Equation.3 funksiyaning EMBED Equation.3 nuqtadagi hosilasini topamiz. Buning uchun EMBED Equation.3 nuqtada EMBED Equation.3 argumentga EMBED Equation.3 orttirma beramiz va funksiyaning mos orttirmasini topamiz: EMBED Equation.3
EMBED Equation.3 .
Orttirmalar nisbatini tuzamiz:
EMBED Equation.3 .
Bu nisbatning EMBED Equation.3 dagi limitini topamiz:
EMBED Equation.3 .
2. EMBED Equation.3 funksiyaning hosilasini hosila ta’rifini va tangenslar ayirmasi formulasini qo‘llab, topamiz:
EMBED Equation.3
EMBED Equation.3
2-ta’rif. EMBED Equation.3 funksiyaning EMBED Equation.3 nuqtadagi o‘ng (chap) hosilasi deb
EMBED Equation.3 limitga aytiladi.
Misol. EMBED Equation.3 funksiyaning EMBED Equation.3 nuqtadagi o‘ng va chap hosilalarini topamiz. Berilgan funksiyaning EMBED Equation.3 nuqtadagi orttirmasini topamiz:
EMBED Equation.3
U holda
EMBED Equation.3 EMBED Equation.3
Bu misolda EMBED Equation.3 Shu sababli EMBED Equation.3 funksiya uchun EMBED Equation.3 da EMBED Equation.3 nisbatning limiti mavjud emas va EMBED Equation.3 funksiya EMBED Equation.3 nuqtada hosilaga ega bo‘lmaydi.
Funksiya hosilasining yuqorida keltirilgan ta’riflaridan ushbu tasdiqlar kelib chiqadi: agar funksiya EMBED Equation.3 nuqtada hosilaga ega bo‘lsa, funksiya shu nuqtada bir-biriga teng bo‘lgan o‘ng va chap hosilalarga ega bo‘lib, EMBED Equation.3 bo‘ladi; agar funksiya EMBED Equation.3 nuqtada o‘ng va chap hosilalarga ega bo‘lib, EMBED Equation.3 bo‘lsa, funksiya shu nuqtada hosilaga ega va EMBED Equation.3 bo‘ladi.
Funksiyaning hosilasini topishga funksiyani differensiallash deyiladi.
Agar EMBED Equation.3 funksiya biror oraliqda aniqlangan bo‘lsa va EMBED Equation.3 hosila bu oraliqning har bir nuqtasida mavjud bo‘lsa, u holda
EMBED Equation.3
formula EMBED Equation.3 hosilani EMBED Equation.3 ning funksiyasi sifatida aniqlaydi. Bundan keyin, agar
EMBED Equation.3 funksiyani differensiallashda nuqta ko‘rsatilmagan bo‘lsa, hosilani
EMBED Equation.3 ning mumkin bo‘lgan barcha qiymatlarida topamiz va EMBED Equation.3 deb yozamiz.