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

解

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

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

解答ステップ

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

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

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

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

簡素化

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

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