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

解

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

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

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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