derivative of (2x/(sqrt(x)-1))

解

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

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

解答ステップ

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

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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