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

解

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

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

解答ステップ

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

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

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

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

簡素化

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

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