derivative (-u^2+5)/((u^2+5)^2)

解

導関数

\frac{- u ^{2} + 5}{( u ^{2} + 5 ) ^{2}}

- \frac{2 u ( - u ^{2} + 15 )}{( u ^{2} + 5 ) ^{3}}

解答ステップ

\frac{d}{d u} (\frac{- u ^{2} + 5}{( u ^{2} + 5 ) ^{2}})

商の法則を適用:

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

= \frac{\frac{d}{d u} ( - u ^{2} + 5 ) ( u ^{2} + 5 ) ^{2} - \frac{d}{d u} ( ( u ^{2} + 5 ) ^{2} ) ( - u ^{2} + 5 )}{( ( u ^{2} + 5 ) ^{2} ) ^{2}}

\frac{d}{d u} (- u^{2} + 5) = - 2 u

\frac{d}{d u} ((u^{2} + 5)^{2}) = 4 u (u^{2} + 5)

= \frac{( - 2 u ) ( u ^{2} + 5 ) ^{2} - 4 u ( u ^{2} + 5 ) ( - u ^{2} + 5 )}{( ( u ^{2} + 5 ) ^{2} ) ^{2}}

簡素化

\frac{( - 2 u ) ( u ^{2} + 5 ) ^{2} - 4 u ( u ^{2} + 5 ) ( - u ^{2} + 5 )}{( ( u ^{2} + 5 ) ^{2} ) ^{2}} : - \frac{2 u ( - u ^{2} + 15 )}{( u ^{2} + 5 ) ^{3}}

= - \frac{2 u ( - u ^{2} + 15 )}{( u ^{2} + 5 ) ^{3}}