derivative of (sqrt(x)/(ln(x)))

解

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

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

解答ステップ

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

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

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

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

簡素化

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

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