derivative y=ln(sqrt((x+1)/(x-1)))

解

導関数

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

- \frac{1}{( x - 1 ) ( x + 1 )}

解答ステップ

\frac{d}{d x} (ln (\frac{x + 1}{x - 1}))

連鎖法則を適用:

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

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

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

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

簡素化

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

= - \frac{1}{( x - 1 ) ( x + 1 )}