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

解

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

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

解答ステップ

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

定数を除去:

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

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

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

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

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

簡素化

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

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