derivative of-(4x/((-x+1)^2(x+1)^2))

解

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

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

解答ステップ

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

定数を除去:

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

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

\frac{d x}{d x} = 1

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

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

簡素化

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

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