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

解

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

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

解答ステップ

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

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

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

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

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

簡素化

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

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