derivative y=((x^2+3)/(x^2-3))^4

解

導関数

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

- \frac{48 x ( x ^{2} + 3 ) ^{3}}{( x ^{2} - 3 ) ^{5}}

解答ステップ

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

連鎖法則を適用:

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

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

\frac{d}{d x} (\frac{x ^{2} + 3}{x ^{2} - 3}) = - \frac{12 x}{( x ^{2} - 3 ) ^{2}}

= 4 (\frac{x ^{2} + 3}{x ^{2} - 3})^{3} (- \frac{12 x}{( x ^{2} - 3 ) ^{2}})

簡素化

4 (\frac{x ^{2} + 3}{x ^{2} - 3})^{3} (- \frac{12 x}{( x ^{2} - 3 ) ^{2}}) : - \frac{48 x ( x ^{2} + 3 ) ^{3}}{( x ^{2} - 3 ) ^{5}}

= - \frac{48 x ( x ^{2} + 3 ) ^{3}}{( x ^{2} - 3 ) ^{5}}