derivative of 1/(x(8+ln(x)))

解

\frac{d}{d x} (\frac{1}{x ( 8 + ln ( x ) )})

- \frac{ln ( x ) + 9}{x ^{2} ( 8 + ln ( x ) ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{1}{x ( 8 + ln ( x ) )})

指数の規則を適用する:

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

= \frac{d}{d x} ((x (8 + ln (x)))^{- 1})

連鎖法則を適用:

- \frac{1}{( x ( 8 + ln ( x ) ) ) ^{2}} \frac{d}{d x} (x (8 + ln (x)))

= - \frac{1}{( x ( 8 + ln ( x ) ) ) ^{2}} \frac{d}{d x} (x (8 + ln (x)))

\frac{d}{d x} (x (8 + ln (x))) = ln (x) + 9

= - \frac{1}{( x ( 8 + ln ( x ) ) ) ^{2}} (ln (x) + 9)

簡素化

- \frac{1}{( x ( 8 + ln ( x ) ) ) ^{2}} (ln (x) + 9) : - \frac{ln ( x ) + 9}{x ^{2} ( 8 + ln ( x ) ) ^{2}}

= - \frac{ln ( x ) + 9}{x ^{2} ( 8 + ln ( x ) ) ^{2}}