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

解

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

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

解答ステップ

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

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

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

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

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

簡素化

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

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