derivative (1+sqrt(X))/(1-sqrt(X))

解

導関数

\frac{1 + X}{1 - X}

\frac{1}{X ( 1 - X ) ^{2}}

解答ステップ

\frac{d}{d X} (\frac{1 + X}{1 - X})

商の法則を適用:

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

= \frac{\frac{d}{d X} ( 1 + X ) ( 1 - X ) - \frac{d}{d X} ( 1 - X ) ( 1 + X )}{( 1 - X ) ^{2}}

\frac{d}{d X} (1 + X) = \frac{1}{2 X}

\frac{d}{d X} (1 - X) = - \frac{1}{2 X}

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

簡素化

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

= \frac{1}{X ( 1 - X ) ^{2}}