3.4-bo'lim : Mahsulot va miqdor qoidasi
2. \(y = \left( {1 + \sqrt {{x^3}} } \right)\,\left( {{x^{ - 3}} - 2\sqrt[3]{x}} \right)\) hosilasini topish uchun Mahsulot qoidasidan foydalaning.
Yechim
Bu erda mahsulot qoidasidan foydalanib lotinni olishdan boshqa ko'p ish yo'q. Shuningdek, biz ildizlarni kasr ko'rsatkichlariga aylantirishimiz kerak.
\[y = \left( {1 + {x^{\frac{3}{2}}}} \right)\,\left( {{x^{ - 3}} - 2{x^{\frac{1}{3}}}} \right)\]Demak, hosila,
\[\frac{{dy}}{{dx}} = \left( {\frac{3}{2}{x^{\frac{1}{2}}}} \right)\,\left( {{x^{ - 3}} - 2{x^{\frac{1}{3}}}} \right) + \left( {1 + {x^{\frac{3}{2}}}} \right)\,\left( { - 3{x^{ - 4}} - \frac{2}{3}{x^{ - \,\,\frac{2}{3}}}} \right) = - 3{x^{ - 4}} - \frac{3}{2}{x^{ - \,\,\frac{5}{2}}} - \frac{2}{3}{x^{ - \,\,\frac{2}{3}}} - \frac{{11}}{3}{x^{\frac{5}{6}}}\]E'tibor bering, biz "oddiyroq" javob olish uchun hamma narsani ko'paytirdik.