derivative of (ln(x)/(2x))

解

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

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

解答ステップ

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

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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