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

解

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

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

解答ステップ

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

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

\frac{d x}{d x} = 1

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

簡素化

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

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