derivative f(x)=(x^2-2/x)^{-5}

解

導関数

f (x) = (x^{2} - \frac{2}{x})^{- 5}

- \frac{5 x ^{4} ( 2 x ^{3} + 2 )}{( x ^{3} - 2 ) ^{6}}

解答ステップ

\frac{d}{d x} ((x^{2} - \frac{2}{x})^{- 5})

連鎖法則を適用:

- \frac{5}{( x ^{2} - \frac{2}{x} ) ^{6}} \frac{d}{d x} (x^{2} - \frac{2}{x})

= - \frac{5}{( x ^{2} - \frac{2}{x} ) ^{6}} \frac{d}{d x} (x^{2} - \frac{2}{x})

\frac{d}{d x} (x^{2} - \frac{2}{x}) = 2 x + \frac{2}{x ^{2}}

= - \frac{5}{( x ^{2} - \frac{2}{x} ) ^{6}} (2 x + \frac{2}{x ^{2}})

簡素化

- \frac{5}{( x ^{2} - \frac{2}{x} ) ^{6}} (2 x + \frac{2}{x ^{2}}) : - \frac{5 x ^{4} ( 2 x ^{3} + 2 )}{( x ^{3} - 2 ) ^{6}}

= - \frac{5 x ^{4} ( 2 x ^{3} + 2 )}{( x ^{3} - 2 ) ^{6}}