(\partial)/(\partial y)((x^3+(y/2))^{2/3})

解

\frac{\partial}{\partial y} ((x^{3} + (\frac{y}{2}))^{\frac{2}{3}})

\frac{2 ^{\frac{1}{3}}}{3 ( 2 x ^{3} + y ) ^{\frac{1}{3}}}

解答ステップ

\frac{\partial}{\partial y} ((x^{3} + (\frac{y}{2}))^{\frac{2}{3}})

簡素化

(x^{3} + (\frac{y}{2}))^{\frac{2}{3}} : (\frac{2 x ^{3} + y}{2})^{\frac{2}{3}}

= \frac{\partial}{\partial y} ((\frac{2 x ^{3} + y}{2})^{\frac{2}{3}})

x

を定数として扱う

連鎖法則を適用:

\frac{2}{3 ( \frac{2 x ^{3} + y}{2} ) ^{\frac{1}{3}}} \frac{\partial}{\partial y} (\frac{2 x ^{3} + y}{2})

= \frac{2}{3 ( \frac{2 x ^{3} + y}{2} ) ^{\frac{1}{3}}} \frac{\partial}{\partial y} (\frac{2 x ^{3} + y}{2})

\frac{\partial}{\partial y} (\frac{2 x ^{3} + y}{2}) = \frac{1}{2}

= \frac{2}{3 ( \frac{2 x ^{3} + y}{2} ) ^{\frac{1}{3}}} \cdot \frac{1}{2}

簡素化

\frac{2}{3 ( \frac{2 x ^{3} + y}{2} ) ^{\frac{1}{3}}} \cdot \frac{1}{2} : \frac{2 ^{\frac{1}{3}}}{3 ( 2 x ^{3} + y ) ^{\frac{1}{3}}}

= \frac{2 ^{\frac{1}{3}}}{3 ( 2 x ^{3} + y ) ^{\frac{1}{3}}}