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

解

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

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

解答ステップ

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

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

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

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

簡素化

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

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