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

解

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

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

解答ステップ

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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