Funktsiyaning differentsiallanuvchanligi. Koshi- riman shartlari.
funksiyaning nuqtadagi hosilasi deb, quyidagi limitga aytiladi:
Teorema funksiya nuqtada differensiallanuvchi bo’lishi uchun quyidagi shart bajarilashi yetarli:
Xususiy hosilalar nuqta atrofida mavjud va uzluksiz bo’lishi kerak;
nuqtada Koshi –Riman shartlari bajarilishi kerak:
Hosila quyidagi teng kuchli formulalar orqali topiladi:
Auditoriya topshirig’i
Misol1
funksiyaning xosila mavjud bo’lgan nuqtalarini toping.
Yechish:
Ma`lumki
Ya`ni
Xususiy hosilalarni topamiz va ularni Koshi-Riman shartlarini bajarilishini tekshiramiz
Ya’ni Koshi-Riman shartlari barcha nuqtalar uchun bajariladi, shu hosilani topamiz:
2-misol
funksiya hosilasini toping
Yechish: ![](data:image/png;base64,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)
Ko’rinib turibdiki,
Ya’ni Koshi-Riman shartlari hech qaysi nuqtalar uchun bajarilmadi, demak hosila mavjud emas.
3-misol
funksiya hosilasini toping.
Yechish:
Ya’ni![](data:image/png;base64,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)
Bundan berilgan funksiya ihtiyoriy nuqtada differensiallanuvchi ekanligini ko’rsatadi.
Hosilani topaylik:
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