(\partial)/(\partial z)(\sqrt[3]{xyz})

解

\frac{\partial}{\partial z} (3 x yz)

\frac{x ^{\frac{1}{3}} y ^{\frac{1}{3}}}{3 z ^{\frac{2}{3}}}

解答ステップ

\frac{\partial}{\partial z} (3 x yz)

x, y

を定数として扱う

連鎖法則を適用:

\frac{1}{3 ( x yz ) ^{\frac{2}{3}}} \frac{\partial}{\partial z} (x yz)

= \frac{1}{3 ( x yz ) ^{\frac{2}{3}}} \frac{\partial}{\partial z} (x yz)

\frac{\partial}{\partial z} (x yz) = x y

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

簡素化

\frac{1}{3 ( x yz ) ^{\frac{2}{3}}} x y : \frac{x ^{\frac{1}{3}} y ^{\frac{1}{3}}}{3 z ^{\frac{2}{3}}}

= \frac{x ^{\frac{1}{3}} y ^{\frac{1}{3}}}{3 z ^{\frac{2}{3}}}