derivative u/(u^2+4)

解

導関数

\frac{u}{u ^{2} + 4}

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

解答ステップ

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

商の法則を適用:

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

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

\frac{d u}{d u} = 1

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

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

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

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