derivative of x/(log_{e(x)})

解

\frac{d}{d x} (\frac{x}{lo g _{e} ( x )})

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

解答ステップ

\frac{d}{d x} (\frac{x}{lo g _{e} ( 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} lo g _{e} ( x ) - \frac{d}{d x} ( lo g _{e} ( x ) ) x}{( lo g _{e} ( x ) ) ^{2}}

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

\frac{d}{d x} (lo g_{e} (x)) = \frac{1}{x}

= \frac{1 \cdot lo g _{e} ( x ) - \frac{1}{x} x}{( lo g _{e} ( x ) ) ^{2}}

簡素化

\frac{1 \cdot lo g _{e} ( x ) - \frac{1}{x} x}{lo g _{e} ( x ) ^{2}} : \frac{ln ( x ) - 1}{ln ^{2} ( x )}

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