derivative (5u)/(4u^2+1)

解

導関数

\frac{5 u}{4 u ^{2} + 1}

\frac{5 ( - 4 u ^{2} + 1 )}{( 4 u ^{2} + 1 ) ^{2}}

解答ステップ

\frac{d}{d u} (\frac{5 u}{4 u ^{2} + 1})

定数を除去:

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

= 5 \frac{d}{d u} (\frac{u}{4 u ^{2} + 1})

商の法則を適用:

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

= 5 \cdot \frac{\frac{d u}{d u} ( 4 u ^{2} + 1 ) - \frac{d}{d u} ( 4 u ^{2} + 1 ) u}{( 4 u ^{2} + 1 ) ^{2}}

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

\frac{d}{d u} (4 u^{2} + 1) = 8 u

= 5 \cdot \frac{1 \cdot ( 4 u ^{2} + 1 ) - 8 uu}{( 4 u ^{2} + 1 ) ^{2}}

簡素化

5 \cdot \frac{1 \cdot ( 4 u ^{2} + 1 ) - 8 uu}{( 4 u ^{2} + 1 ) ^{2}} : \frac{5 ( - 4 u ^{2} + 1 )}{( 4 u ^{2} + 1 ) ^{2}}

= \frac{5 ( - 4 u ^{2} + 1 )}{( 4 u ^{2} + 1 ) ^{2}}