derivative of ln(sqrt((x+9/(x-9))))

解

\frac{d}{d x} (ln (\frac{x + 9}{x - 9}))

- \frac{9}{( x - 9 ) ( x + 9 )}

解答ステップ

\frac{d}{d x} (ln (\frac{x + 9}{x - 9}))

連鎖法則を適用:

\frac{1}{\frac{x + 9}{x - 9}} \frac{d}{d x} (\frac{x + 9}{x - 9})

= \frac{1}{\frac{x + 9}{x - 9}} \frac{d}{d x} (\frac{x + 9}{x - 9})

\frac{d}{d x} (\frac{x + 9}{x - 9}) = - \frac{9}{x + 9 ( x - 9 ) ^{\frac{3}{2}}}

= \frac{1}{\frac{x + 9}{x - 9}} (- \frac{9}{x + 9 ( x - 9 ) ^{\frac{3}{2}}})

簡素化

\frac{1}{\frac{x + 9}{x - 9}} (- \frac{9}{x + 9 ( x - 9 ) ^{\frac{3}{2}}}) : - \frac{9}{( x - 9 ) ( x + 9 )}

= - \frac{9}{( x - 9 ) ( x + 9 )}