derivative y=(((x^2-2)/(2x^2+1))^2)

解

導関数

y = ((\frac{x ^{2} - 2}{2 x ^{2} + 1})^{2})

\frac{20 x ( x ^{2} - 2 )}{( 2 x ^{2} + 1 ) ^{3}}

解答ステップ

\frac{d}{d x} ((\frac{x ^{2} - 2}{2 x ^{2} + 1})^{2})

連鎖法則を適用:

2 \cdot \frac{x ^{2} - 2}{2 x ^{2} + 1} \frac{d}{d x} (\frac{x ^{2} - 2}{2 x ^{2} + 1})

= 2 \cdot \frac{x ^{2} - 2}{2 x ^{2} + 1} \frac{d}{d x} (\frac{x ^{2} - 2}{2 x ^{2} + 1})

\frac{d}{d x} (\frac{x ^{2} - 2}{2 x ^{2} + 1}) = \frac{10 x}{( 2 x ^{2} + 1 ) ^{2}}

= 2 \cdot \frac{x ^{2} - 2}{2 x ^{2} + 1} \cdot \frac{10 x}{( 2 x ^{2} + 1 ) ^{2}}

簡素化

2 \cdot \frac{x ^{2} - 2}{2 x ^{2} + 1} \cdot \frac{10 x}{( 2 x ^{2} + 1 ) ^{2}} : \frac{20 x ( x ^{2} - 2 )}{( 2 x ^{2} + 1 ) ^{3}}

= \frac{20 x ( x ^{2} - 2 )}{( 2 x ^{2} + 1 ) ^{3}}