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

解

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

- \frac{8 x ( - x ^{2} - 12 )}{( x ^{2} - 4 ) ^{3}}

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

= 4 \cdot \frac{\frac{d}{d x} ( - x ^{2} - 4 ) ( x ^{2} - 4 ) ^{2} - \frac{d}{d x} ( ( x ^{2} - 4 ) ^{2} ) ( - x ^{2} - 4 )}{( ( x ^{2} - 4 ) ^{2} ) ^{2}}

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

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

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

簡素化

4 \cdot \frac{( - 2 x ) ( x ^{2} - 4 ) ^{2} - 4 x ( x ^{2} - 4 ) ( - x ^{2} - 4 )}{( ( x ^{2} - 4 ) ^{2} ) ^{2}} : - \frac{8 x ( - x ^{2} - 12 )}{( x ^{2} - 4 ) ^{3}}

= - \frac{8 x ( - x ^{2} - 12 )}{( x ^{2} - 4 ) ^{3}}