derivative (4u^2)/((u^2+u)^3)

解

導関数

\frac{4 u ^{2}}{( u ^{2} + u ) ^{3}}

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

解答ステップ

\frac{d}{d u} (\frac{4 u ^{2}}{( u ^{2} + u ) ^{3}})

定数を除去:

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

= 4 \frac{d}{d u} (\frac{u ^{2}}{( u ^{2} + u ) ^{3}})

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

\frac{d}{d u} (u^{2}) = 2 u

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

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

簡素化

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

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