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Differensiallah qoidalri va formulalari
Yig‘indi, ayirma, ko‘paytma va bo‘linmani differensiallash
Funksiyaning hosilasi ta’rifidan foydalanib ikki funksiya yig‘indisi, ayirmasi, ko‘paytmasi va bo‘linmasini differensiallash qoidalarini keltirib chiqaramiz.
3-teorema. Agar EMBED Equation.3 va EMBED Equation.3 funksiyalar EMBED Equation.3 nuqtada differensiallanuvchi bo‘lsa, u holda bu funksiyalarning yig‘indisi, ayirmasi, ko‘paytmasi va bo‘linmasi (bo‘linmasi EMBED Equation.3 shart bajarilganda) ham EMBED Equation.3 nuqtada differensiallanuvchi va quyidagi formulalar o‘rinli bo‘ladi:
1. EMBED Equation.3 ; 2. EMBED Equation.3 3. EMBED Equation.3 .
Asosiy elementar funksiyalarning hosilalari
Asosiy elementar funksiyalarning hosilalarini topishda 17-§ da keltirilgan ekvivalent cheksiz kichik funksiyalardan, teskari va murakkab funksiyalarni differensiallash formulalaridan hamda yig‘indi, ayirma, ko‘paytma va bo‘linmani differensiallash qoidalaridan foydalanamiz.
1. O‘zgarmas funksiya: EMBED Equation.3 ( EMBED Equation.3 ). O‘garmas funksiya butun sonlar o‘qida o‘zgarmas qiymatini saqlagani uchun ixtiyoriy nuqtada uning orttirmasi nolga teng bo‘ladi. Shu sababli
EMBED Equation.3
2. Darajali funksiya: EMBED Equation.3 , bunda EMBED Equation.3 . Bu funksiya uchun EMBED Equation.3 da
EMBED Equation.3
bo‘ladi.
Bundan
EMBED Equation.3
EMBED Equation.3 da EMBED Equation.3 ~ EMBED Equation.3 ni hisobga olib, topamiz:
EMBED Equation.3
Demak,
EMBED Equation.3
Xususan, EMBED Equation.3 EMBED Equation.3
3. Ko‪rsatkichli funksiya: EMBED Equation.3 bunda EMBED Equation.3 . Bu funksiyaning orttirmasi EMBED Equation.3 ga teng bo‘lib, EMBED Equation.3 bo‘ladi.‬‬‬‬‬
Bundan EMBED Equation.3 da EMBED Equation.3  EMBED Equation.3 ni hisobga olib, topamiz:
EMBED Equation.3
Demak, EMBED Equation.3 Xususan, EMBED Equation.3