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

解

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

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

解答ステップ

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

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

\frac{d}{d x} ((x^{2} + 1)^{2}) = 4 x (x^{2} + 1)

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

簡素化

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

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