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

解

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

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

解答ステップ

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

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

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

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

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

簡素化

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