derivative of (2-5x-x^2/(x^2-6))

解

\frac{d}{d x} (\frac{2 - 5 x - x ^{2}}{x ^{2} - 6})

\frac{5 x ^{2} + 8 x + 30}{( x ^{2} - 6 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{2 - 5 x - x ^{2}}{x ^{2} - 6})

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 - 5 x - x ^{2} ) ( x ^{2} - 6 ) - \frac{d}{d x} ( x ^{2} - 6 ) ( 2 - 5 x - x ^{2} )}{( x ^{2} - 6 ) ^{2}}

\frac{d}{d x} (2 - 5 x - x^{2}) = - 2 x - 5

\frac{d}{d x} (x^{2} - 6) = 2 x

= \frac{( - 2 x - 5 ) ( x ^{2} - 6 ) - 2 x ( 2 - 5 x - x ^{2} )}{( x ^{2} - 6 ) ^{2}}

拡張

(- 2 x - 5) (x^{2} - 6) - 2 x (2 - 5 x - x^{2}) : 5 x^{2} + 8 x + 30

= \frac{5 x ^{2} + 8 x + 30}{( x ^{2} - 6 ) ^{2}}