3.10-bo'lim : Yashirin farqlash
2. \({x^2} + {y^3} = 4\) uchun quyidagi amallarning har birini bajaring.
- Y uchun tenglamani yechish va to'g'ridan-to'g'ri farqlash orqali \(y'\) ni toping.
- Yashirin farqlash orqali \(y'\) ni toping.
- (a) va (b) dagi hosilalar bir xil ekanligini tekshiring.
Yechim
Birinchidan, biz faqat \(y\) uchun tenglamani hal qilishimiz kerak.
\[{y^3} = 4 - {x^2}\hspace{0.25in} \Rightarrow \hspace{0.25in}\,y = {\left( {4 - {x^2}} \right)^{\frac{1}{3}}}\]2-qadam
Endi \(x\) ga nisbatan farqlang.
\[\require{bbox} \bbox[2pt,border:1px solid black]{{y' = - {\textstyle{2 \over 3}}x{{\left( {4 - {x^2}} \right)}^{ - \,\,\frac{2}{3}}}}}\]Yechim
Birinchidan, biz \(x\)ga nisbatan hamma narsaning hosilasini olishimiz kerak va biz \(y\) haqiqatan ham \(y\left( x \right)\) ekanligini esga olishimiz kerak, shuning uchun biz \(y\) ishtirokidagi atamalarning hosilasini olishda zanjir qoidasidan foydalanishimiz kerak.
Hosilini olish, beradi,
\[2x + 3{y^2}y' = 0\]2-qadam
Va nihoyat, biz qilishimiz kerak bo'lgan yagona narsa buni \(y'\) uchun hal qilishdir.
\[y' = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{{2x}}{{3{y^2}}}}}\]c (a) va (b) dagi hosilalar bir xil ekanligini tekshiring. Yechim
Kimdan (a) bizda \(x\) funksiyasi sifatida aniq yozilgan \(y\) formulasi bor, shuning uchun uni biz topgan lotinga ulang. (b) va biroz soddalashtirish/ishlash bilan biz kiritganimizdek hosila olishimizni ko'rsating (a).
\[y' = - \frac{{2x}}{{3{y^2}}} = - \frac{{2x}}{{3{{\left( {4 - {x^2}} \right)}^{\frac{2}{3}}}}} = - {\textstyle{2 \over 3}}x{\left( {4 - {x^2}} \right)^{ - \,\,\frac{2}{3}}}\]Shunday qilib, biz xuddi shunday hosila oldik.