(\partial)/(\partial y)((x+y)/(xy-1))

解

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

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

解答ステップ

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

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

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

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

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

簡素化

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

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