derivative x/(2+x^2)

解

導関数

\frac{x}{2 + x ^{2}}

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

解答ステップ

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

商の法則を適用:

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

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

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

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

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

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

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