derivative of (x^{1/3}+x/(ln(x)+x))

解

\frac{d}{d x} (\frac{x ^{\frac{1}{3}} + x}{ln ( x ) + x})

\frac{- 2 x + 3 x ^{\frac{2}{3}} ln ( x ) + ln ( x ) - 3 - 3 x ^{\frac{2}{3}}}{3 x ^{\frac{2}{3}} ( ln ( x ) + x ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{\frac{1}{3}} + x}{ln ( x ) + x})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

簡素化

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

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