derivative (9-ln(x))/(ln(x)-1)

解

導関数

\frac{9 - ln ( x )}{ln ( x ) - 1}

- \frac{8}{x ( ln ( x ) - 1 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{9 - ln ( x )}{ln ( x ) - 1})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( 9 - ln ( x ) ) ( ln ( x ) - 1 ) - \frac{d}{d x} ( ln ( x ) - 1 ) ( 9 - ln ( x ) )}{( ln ( x ) - 1 ) ^{2}}

\frac{d}{d x} (9 - ln (x)) = - \frac{1}{x}

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

= \frac{( - \frac{1}{x} ) ( ln ( x ) - 1 ) - \frac{1}{x} ( 9 - ln ( x ) )}{( ln ( x ) - 1 ) ^{2}}

簡素化

\frac{( - \frac{1}{x} ) ( ln ( x ) - 1 ) - \frac{1}{x} ( 9 - ln ( x ) )}{( ln ( x ) - 1 ) ^{2}} : - \frac{8}{x ( ln ( x ) - 1 ) ^{2}}

= - \frac{8}{x ( ln ( x ) - 1 ) ^{2}}