derivative of ln((5x+1/(5x-1)))

解

\frac{d}{d x} (ln (\frac{5 x + 1}{5 x - 1}))

- \frac{10}{( 5 x + 1 ) ( 5 x - 1 )}

解答ステップ

\frac{d}{d x} (ln (\frac{5 x + 1}{5 x - 1}))

連鎖法則を適用:

\frac{1}{\frac{5 x + 1}{5 x - 1}} \frac{d}{d x} (\frac{5 x + 1}{5 x - 1})

= \frac{1}{\frac{5 x + 1}{5 x - 1}} \frac{d}{d x} (\frac{5 x + 1}{5 x - 1})

\frac{d}{d x} (\frac{5 x + 1}{5 x - 1}) = - \frac{10}{( 5 x - 1 ) ^{2}}

= \frac{1}{\frac{5 x + 1}{5 x - 1}} (- \frac{10}{( 5 x - 1 ) ^{2}})

簡素化

\frac{1}{\frac{5 x + 1}{5 x - 1}} (- \frac{10}{( 5 x - 1 ) ^{2}}) : - \frac{10}{( 5 x + 1 ) ( 5 x - 1 )}

= - \frac{10}{( 5 x + 1 ) ( 5 x - 1 )}