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

解

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

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

解答ステップ

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

x

を定数として扱う

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

\frac{\partial}{\partial y} (x) = 0

= - \frac{2 x y}{( x ^{2} + y ^{2} ) ^{2}} + 0

簡素化

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