derivative 1/(\sqrt[3]{(8-x^3)^8)}

解

導関数

\frac{1}{3 ( 8 - x ^{3} ) ^{8}}

\frac{8 x ^{2} ( - x ^{3} + 8 ) ^{7}}{( ( 8 - x ^{3} ) ^{8} ) ^{\frac{4}{3}}}

解答ステップ

\frac{d}{d x} \frac{1}{3 ( 8 - x ^{3} ) ^{8}}

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

= \frac{d}{d x} (((8 - x^{3})^{8})^{- \frac{1}{3}})

連鎖法則を適用:

- \frac{1}{3 ( ( 8 - x ^{3} ) ^{8} ) ^{\frac{4}{3}}} \frac{d}{d x} ((8 - x^{3})^{8})

= - \frac{1}{3 ( ( 8 - x ^{3} ) ^{8} ) ^{\frac{4}{3}}} \frac{d}{d x} ((8 - x^{3})^{8})

\frac{d}{d x} ((8 - x^{3})^{8}) = - 24 x^{2} (8 - x^{3})^{7}

= - \frac{1}{3 ( ( 8 - x ^{3} ) ^{8} ) ^{\frac{4}{3}}} (- 24 x^{2} (8 - x^{3})^{7})

簡素化

- \frac{1}{3 ( ( 8 - x ^{3} ) ^{8} ) ^{\frac{4}{3}}} (- 24 x^{2} (8 - x^{3})^{7}) : \frac{8 x ^{2} ( - x ^{3} + 8 ) ^{7}}{( ( 8 - x ^{3} ) ^{8} ) ^{\frac{4}{3}}}

= \frac{8 x ^{2} ( - x ^{3} + 8 ) ^{7}}{( ( 8 - x ^{3} ) ^{8} ) ^{\frac{4}{3}}}