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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Sum/Difference Rule:

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

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

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

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

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

Simplify

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

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