derivative y=((x^2+4)/(x^2-4))^4

解

導関数

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

- \frac{64 x ( x ^{2} + 4 ) ^{3}}{( x ^{2} - 4 ) ^{5}}

解答ステップ

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

連鎖法則を適用:

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

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

\frac{d}{d x} (\frac{x ^{2} + 4}{x ^{2} - 4}) = - \frac{16 x}{( x ^{2} - 4 ) ^{2}}

= 4 (\frac{x ^{2} + 4}{x ^{2} - 4})^{3} (- \frac{16 x}{( x ^{2} - 4 ) ^{2}})

簡素化

4 (\frac{x ^{2} + 4}{x ^{2} - 4})^{3} (- \frac{16 x}{( x ^{2} - 4 ) ^{2}}) : - \frac{64 x ( x ^{2} + 4 ) ^{3}}{( x ^{2} - 4 ) ^{5}}

= - \frac{64 x ( x ^{2} + 4 ) ^{3}}{( x ^{2} - 4 ) ^{5}}