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

解

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

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

解答ステップ

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

定数を除去:

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

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

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

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

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

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

簡素化

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

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