derivative of (x-2/(x^2+5x+4))

解

\frac{d}{d x} (\frac{x - 2}{x ^{2} + 5 x + 4})

\frac{- x ^{2} + 4 x + 14}{( x ^{2} + 5 x + 4 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x - 2}{x ^{2} + 5 x + 4})

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 ^{2} + 5 x + 4 ) - \frac{d}{d x} ( x ^{2} + 5 x + 4 ) ( x - 2 )}{( x ^{2} + 5 x + 4 ) ^{2}}

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

\frac{d}{d x} (x^{2} + 5 x + 4) = 2 x + 5

= \frac{1 \cdot ( x ^{2} + 5 x + 4 ) - ( 2 x + 5 ) ( x - 2 )}{( x ^{2} + 5 x + 4 ) ^{2}}

簡素化

\frac{1 \cdot ( x ^{2} + 5 x + 4 ) - ( 2 x + 5 ) ( x - 2 )}{( x ^{2} + 5 x + 4 ) ^{2}} : \frac{- x ^{2} + 4 x + 14}{( x ^{2} + 5 x + 4 ) ^{2}}

= \frac{- x ^{2} + 4 x + 14}{( x ^{2} + 5 x + 4 ) ^{2}}