(\partial)/(\partial u)(b((((Nu))/(V(u)))^{(2/3)}))

解

\frac{\partial}{\partial u} (b ((\frac{( N u )}{V ( u )})^{(\frac{2}{3})}))

\frac{2 b N ^{\frac{2}{3}} ( - u \frac{\partial}{\partial u} ( V ( u ) ) + V ( u ) )}{3 u ^{\frac{1}{3}} V ( u ) ^{\frac{5}{3}}}

解答ステップ

\frac{\partial}{\partial u} (b ((\frac{( N u )}{V ( u )})^{(\frac{2}{3})}))

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= b \frac{\partial}{\partial u} ((\frac{N u}{V ( u )})^{\frac{2}{3}})

連鎖法則を適用:

\frac{2}{3 ( \frac{N u}{V ( u )} ) ^{\frac{1}{3}}} \frac{\partial}{\partial u} (\frac{N u}{V ( u )})

= \frac{2}{3 ( \frac{N u}{V ( u )} ) ^{\frac{1}{3}}} \frac{\partial}{\partial u} (\frac{N u}{V ( u )})

\frac{\partial}{\partial u} (\frac{N u}{V ( u )}) = \frac{N ( V ( u ) - u \frac{\partial}{\partial u} ( V ( u ) ) )}{V ( u ) ^{2}}

= b \frac{2}{3 ( \frac{N u}{V ( u )} ) ^{\frac{1}{3}}} \cdot \frac{N ( V ( u ) - u \frac{\partial}{\partial u} ( V ( u ) ) )}{V ( u ) ^{2}}

簡素化

b \frac{2}{3 ( \frac{N u}{V ( u )} ) ^{\frac{1}{3}}} \cdot \frac{N ( V ( u ) - u \frac{\partial}{\partial u} ( V ( u ) ) )}{V ( u ) ^{2}} : \frac{2 b N ^{\frac{2}{3}} ( - u \frac{\partial}{\partial u} ( V ( u ) ) + V ( u ) )}{3 u ^{\frac{1}{3}} V ( u ) ^{\frac{5}{3}}}

= \frac{2 b N ^{\frac{2}{3}} ( - u \frac{\partial}{\partial u} ( V ( u ) ) + V ( u ) )}{3 u ^{\frac{1}{3}} V ( u ) ^{\frac{5}{3}}}