derivative (x+2)/(x^3+x-6)

解

導関数

\frac{x + 2}{x ^{3} + x - 6}

\frac{- 2 x ^{3} - 6 x ^{2} - 8}{( x ^{3} + x - 6 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x + 2}{x ^{3} + x - 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} ( x + 2 ) ( x ^{3} + x - 6 ) - \frac{d}{d x} ( x ^{3} + x - 6 ) ( x + 2 )}{( x ^{3} + x - 6 ) ^{2}}

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

\frac{d}{d x} (x^{3} + x - 6) = 3 x^{2} + 1

= \frac{1 \cdot ( x ^{3} + x - 6 ) - ( 3 x ^{2} + 1 ) ( x + 2 )}{( x ^{3} + x - 6 ) ^{2}}

簡素化

\frac{1 \cdot ( x ^{3} + x - 6 ) - ( 3 x ^{2} + 1 ) ( x + 2 )}{( x ^{3} + x - 6 ) ^{2}} : \frac{- 2 x ^{3} - 6 x ^{2} - 8}{( x ^{3} + x - 6 ) ^{2}}

= \frac{- 2 x ^{3} - 6 x ^{2} - 8}{( x ^{3} + x - 6 ) ^{2}}