derivative of (-2(-3x^2+1)/((1+x^2)^3))

解

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

\frac{24 x ( - x ^{2} + 1 )}{( x ^{2} + 1 ) ^{4}}

解答ステップ

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

簡素化

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

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

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

簡素化

- 2 \cdot \frac{( - 6 x ) ( 1 + x ^{2} ) ^{3} - 6 x ( 1 + x ^{2} ) ^{2} ( - 3 x ^{2} + 1 )}{( ( 1 + x ^{2} ) ^{3} ) ^{2}} : \frac{24 x ( - x ^{2} + 1 )}{( x ^{2} + 1 ) ^{4}}

= \frac{24 x ( - x ^{2} + 1 )}{( x ^{2} + 1 ) ^{4}}