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

解

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

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

解答ステップ

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

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

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

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

簡素化

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

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