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

解

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

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

解答ステップ

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

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

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

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

簡素化

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

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