derivative of (sqrt(x)/(2-x))

解

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

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

解答ステップ

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

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

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

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

簡素化

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

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