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

解

導関数

\frac{x ^{2} - 1}{x ^{2} + 1}

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

解答ステップ

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

連鎖法則を適用:

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

= \frac{1}{2 \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}}

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

簡素化

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

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