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

解

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

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

解答ステップ

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

y

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

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

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

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

簡素化

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

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