derivative ((x^2+8)/(x^2-8))^3

解

導関数

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

- \frac{96 x ( x ^{2} + 8 ) ^{2}}{( x ^{2} - 8 ) ^{4}}

解答ステップ

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

連鎖法則を適用:

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

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

\frac{d}{d x} (\frac{x ^{2} + 8}{x ^{2} - 8}) = - \frac{32 x}{( x ^{2} - 8 ) ^{2}}

= 3 (\frac{x ^{2} + 8}{x ^{2} - 8})^{2} (- \frac{32 x}{( x ^{2} - 8 ) ^{2}})

簡素化

3 (\frac{x ^{2} + 8}{x ^{2} - 8})^{2} (- \frac{32 x}{( x ^{2} - 8 ) ^{2}}) : - \frac{96 x ( x ^{2} + 8 ) ^{2}}{( x ^{2} - 8 ) ^{4}}

= - \frac{96 x ( x ^{2} + 8 ) ^{2}}{( x ^{2} - 8 ) ^{4}}