What is (\partial)/(\partial y)(x/(x^2+y^2)+x) ?

The answer to (\partial)/(\partial y)(x/(x^2+y^2)+x) is -(2xy)/((x^2+y^2)^2)

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the Sum/Difference Rule:

(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

Simplify

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

What is (\partial)/(\partial y)(x/(x^2+y^2)+x) ?
The answer to (\partial)/(\partial y)(x/(x^2+y^2)+x) is -(2xy)/((x^2+y^2)^2)