derivative (1-3x-x^2)/(x^2-3)

解

導関数

\frac{1 - 3 x - x ^{2}}{x ^{2} - 3}

\frac{3 x ^{2} + 4 x + 9}{( x ^{2} - 3 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{1 - 3 x - x ^{2}}{x ^{2} - 3})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( 1 - 3 x - x ^{2} ) ( x ^{2} - 3 ) - \frac{d}{d x} ( x ^{2} - 3 ) ( 1 - 3 x - x ^{2} )}{( x ^{2} - 3 ) ^{2}}

\frac{d}{d x} (1 - 3 x - x^{2}) = - 2 x - 3

\frac{d}{d x} (x^{2} - 3) = 2 x

= \frac{( - 2 x - 3 ) ( x ^{2} - 3 ) - 2 x ( 1 - 3 x - x ^{2} )}{( x ^{2} - 3 ) ^{2}}

拡張

(- 2 x - 3) (x^{2} - 3) - 2 x (1 - 3 x - x^{2}) : 3 x^{2} + 4 x + 9

= \frac{3 x ^{2} + 4 x + 9}{( x ^{2} - 3 ) ^{2}}