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

解

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

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

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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