derivative f(x)= x/(1-ln(x-1))

解

導関数

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

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

解答ステップ

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

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

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

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

簡素化

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

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