3.1-bo'lim : Hosilaning ta'rifi
9. ning hosilasini topish uchun hosila ta’rifidan foydalaning,
\[V\left( t \right) = \frac{{t + 1}}{{t + 4}}\]Yechim
Avval biz funktsiyani lotin ta'rifiga ulashimiz kerak.
\[V'\left( t \right) = \mathop {\lim }\limits_{h \to 0} \frac{{V\left( {t + h} \right) - V\left( t \right)}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{1}{h}\left( {\frac{{t + h + 1}}{{t + h + 4}} - \frac{{t + 1}}{{t + 4}}} \right)\]Birinchi funktsiyani baholashni to'g'ri baholaganingizga ishonch hosil qiling. Bu o'quvchilar ushbu muammolarni hal qilishda eng ko'p uchraydigan xatolardan biridir.
Shuni ham yodda tutingki, muammoni o'qishni biroz osonlashtirish uchun ta'rifdagi oqilona ifodani biroz qayta yozing. Buni qilish shart emas, lekin narsalarni ko'rishni biroz chiroyli qiladi.
2-qadam
Keyin ikkita ratsional ifodani bitta ratsional ifodaga birlashtirishimiz kerak.
\[V'\left( t \right) = \mathop {\lim }\limits_{h \to 0} \frac{1}{h}\left( {\frac{{\left( {t + h + 1} \right)\left( {t + 4} \right) - \left( {t + 1} \right)\left( {t + h + 4} \right)}}{{\left( {t + h + 4} \right)\left( {t + 4} \right)}}} \right)\]3-qadam
Endi biz qilishimiz kerak bo'lgan narsa - bu algebra (va bu erda biroz chalkash bo'ladi, lekin bu bunday turdagi muammolar uchun odatiy hol) va biz tugatamiz.
\[\begin{align*}V'\left( t \right) & = \mathop {\lim }\limits_{h \to 0} \frac{1}{h}\left( {\frac{{{t^2} + th + 5t + 4h + 4 - \left( {{t^2} + th + 5t + h + 4} \right)}}{{\left( {t + h + 4} \right)\left( {t + 4} \right)}}} \right)\\ & = \mathop {\lim }\limits_{h \to 0} \frac{1}{h}\left( {\frac{{{t^2} + th + 5t + 4h + 4 - {t^2} - th - 5t - h - 4}}{{\left( {t + h + 4} \right)\left( {t + 4} \right)}}} \right)\\ & = \mathop {\lim }\limits_{h \to 0} \frac{1}{h}\left( {\frac{{3h}}{{\left( {t + h + 4} \right)\left( {t + 4} \right)}}} \right) = \mathop {\lim }\limits_{h \to 0} \frac{3}{{\left( {t + h + 4} \right)\left( {t + 4} \right)}} = \frac{3}{{{{\left( {t + 4} \right)}^2}}}\end{align*}\]Bu funktsiyaning hosilasi u holda,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{V'\left( t \right) = \frac{3}{{{{\left( {t + 4} \right)}^2}}}}}\]