derivative of x/(ln(2x))

解

\frac{d}{d x} (\frac{x}{ln ( 2 x )})

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

解答ステップ

\frac{d}{d x} (\frac{x}{ln ( 2 x )})

商の法則を適用:

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

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

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

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

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

簡素化

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

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