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

解

導関数

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

- \frac{1}{4 ( x ^{2} - x ) x ^{2} - x}

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

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

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

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

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

簡素化

\frac{1}{2} \cdot \frac{2 x ^{2} - x - \frac{2 x - 1}{2 x ^{2} - x} ( 2 x - 1 )}{( x ^{2} - x ) ^{2}} : - \frac{1}{4 ( x ^{2} - x ) x ^{2} - x}

= - \frac{1}{4 ( x ^{2} - x ) x ^{2} - x}