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

解

導関数

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

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

解答ステップ

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

定数を除去:

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

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

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

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

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

簡素化

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

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