(\partial)/(\partial y)(y/(y+x))

解

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

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

解答ステップ

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

x

を定数として扱う

商の法則を適用:

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

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

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

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

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

1 \cdot (y + x) - 1 \cdot y = x

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