3.9-bo'lim : Zanjir qoidasi
27. \(f\left( x \right) = {\left( {\sqrt[3]{{12x}} + {{\sin }^2}\left( {3x} \right)} \right)^{ - 1}}\) ni farqlang.
Maslahat: Ba'zan biz hosilani olishni tugatishimizdan oldin zanjir qoidasini bir necha marta bajarish kerak bo'ladi.
Yechim
Ushbu muammo zanjir qoidasidan bir necha marta foydalanishni talab qiladi va shuning uchun biz har bir foydalanishni aniq qilish uchun hosila jarayoniga qadam qo'yamiz.
Mana lotinning birinchi bosqichi va biz bu bosqichda zanjir qoidasidan foydalanishimiz kerak.
\[f'\left( x \right) = - {\left( {\sqrt[3]{{12x}} + {{\sin }^2}\left( {3x} \right)} \right)^{ - 2}}\frac{d}{{dx}}\left( {{{\left( {12x} \right)}^{\frac{1}{3}}} + {{\sin }^2}\left( {3x} \right)} \right)\]2-qadam
Ko'rib turganimizdek, oldingi bosqichdan olingan hosila har bir shartda zanjir qoidasini ham talab qiladi.
Ushbu bosqichning hosilasi,
\[f'\left( x \right) = - {\left( {\sqrt[3]{{12x}} + {{\sin }^2}\left( {3x} \right)} \right)^{ - 2}}\left( {\frac{1}{3}{{\left( {12x} \right)}^{ - \,\,\frac{2}{3}}}\left( {12} \right) + 2\sin \left( {3x} \right)\frac{d}{{dx}}\left( {\sin \left( {3x} \right)} \right)} \right)\]3-qadam
Ko'rib turganimizdek, ikkinchi muddat yana zanjir qoidasidan foydalanadi.
Demak, hosila,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{f'\left( x \right) = - {{\left( {\sqrt[3]{{12x}} + {{\sin }^2}\left( {3x} \right)} \right)}^{ - 2}}\left( {4{{\left( {12x} \right)}^{ - \,\,\frac{2}{3}}} + 6\sin \left( {3x} \right)\cos \left( {3x} \right)} \right)}}\]