derivative 1/(ln(x+1))

解

導関数

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

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

解答ステップ

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

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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