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

解

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

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

解答ステップ

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

定数を除去:

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

= 5 \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}}

= 5 \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

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

簡素化

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

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