derivative of (a^{2/3}-x^{2/3}^{3/2})

解

\frac{d}{d x} ((a^{\frac{2}{3}} - x^{\frac{2}{3}})^{\frac{3}{2}})

- \frac{( a ^{\frac{2}{3}} - x ^{\frac{2}{3}} ) ^{\frac{1}{2}}}{x ^{\frac{1}{3}}}

解答ステップ

\frac{d}{d x} ((a^{\frac{2}{3}} - x^{\frac{2}{3}})^{\frac{3}{2}})

連鎖法則を適用:

\frac{3 ( a ^{\frac{2}{3}} - x ^{\frac{2}{3}} ) ^{\frac{1}{2}}}{2} \frac{d}{d x} (a^{\frac{2}{3}} - x^{\frac{2}{3}})

= \frac{3 ( a ^{\frac{2}{3}} - x ^{\frac{2}{3}} ) ^{\frac{1}{2}}}{2} \frac{d}{d x} (a^{\frac{2}{3}} - x^{\frac{2}{3}})

\frac{d}{d x} (a^{\frac{2}{3}} - x^{\frac{2}{3}}) = - \frac{2}{3 x ^{\frac{1}{3}}}

= \frac{3 ( a ^{\frac{2}{3}} - x ^{\frac{2}{3}} ) ^{\frac{1}{2}}}{2} (- \frac{2}{3 x ^{\frac{1}{3}}})

簡素化

\frac{3 ( a ^{\frac{2}{3}} - x ^{\frac{2}{3}} ) ^{\frac{1}{2}}}{2} (- \frac{2}{3 x ^{\frac{1}{3}}}) : - \frac{( a ^{\frac{2}{3}} - x ^{\frac{2}{3}} ) ^{\frac{1}{2}}}{x ^{\frac{1}{3}}}

= - \frac{( a ^{\frac{2}{3}} - x ^{\frac{2}{3}} ) ^{\frac{1}{2}}}{x ^{\frac{1}{3}}}