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

解

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

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

解答ステップ

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

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

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

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

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

簡素化

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

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