(\partial)/(\partial x)(x/(1+x^2))

解

\frac{\partial}{\partial x} (\frac{x}{1 + x ^{2}})

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

解答ステップ

\frac{\partial}{\partial 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}}

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

\frac{\partial}{\partial x} (x) = 1

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

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

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

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