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

解

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

\frac{- y + x}{( x + y ) ^{3}}

解答ステップ

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

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

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

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

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

簡素化

\frac{1 \cdot ( x + y ) ^{2} - 2 ( x + y ) y}{( ( x + y ) ^{2} ) ^{2}} : \frac{- y + x}{( x + y ) ^{3}}

= \frac{- y + x}{( x + y ) ^{3}}