3.1-bo'lim : Hosilaning ta'rifi
3. Hosila ta’rifidan foydalanib, funksiyaning hosilasini toping:
\[g\left( x \right) = {x^2}\]Yechim
Avval funksiyani hosila ta'rifiga qo'yishimiz kerak.
\[g'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{g\left( {x + h} \right) - g\left( x \right)}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{{{\left( {x + h} \right)}^2} - {x^2}}}{h}\]Funksiyaning birinchi qiymatini to'g'ri hisoblaganingizga ishonch hosil qiling. Bu talabalar bunday masalalarni yechishda eng ko'p yo'l qo'yadigan xatolardan biridir.
2-qadam
Endi tezkor algebraik hisoblashlarni bajaramiz va masala yechiladi.
\[g'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{{x^2} + 2xh + {h^2} - {x^2}}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{h\left( {2x + h} \right)}}{h} = \mathop {\lim }\limits_{h \to 0} \left( {2x + h} \right) = 2x\]Demak, ushbu funksiyaning hosilasi quyidagicha:
\[\require{bbox} \bbox[2pt,border:1px solid black]{{g'\left( x \right) = 2x}}\]