derivative of ((x^2-1/(x^2+1))^2)

解

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

\frac{8 x ( x ^{2} - 1 )}{( x ^{2} + 1 ) ^{3}}

解答ステップ

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

連鎖法則を適用:

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

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

\frac{d}{d x} (\frac{x ^{2} - 1}{x ^{2} + 1}) = \frac{4 x}{( x ^{2} + 1 ) ^{2}}

= 2 \cdot \frac{x ^{2} - 1}{x ^{2} + 1} \cdot \frac{4 x}{( x ^{2} + 1 ) ^{2}}

簡素化

2 \cdot \frac{x ^{2} - 1}{x ^{2} + 1} \cdot \frac{4 x}{( x ^{2} + 1 ) ^{2}} : \frac{8 x ( x ^{2} - 1 )}{( x ^{2} + 1 ) ^{3}}

= \frac{8 x ( x ^{2} - 1 )}{( x ^{2} + 1 ) ^{3}}