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

The answer to (\partial)/(\partial y)(x/((x^2+y^2)^2)) is -(4xy)/((x^2+y^2)^3)

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

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

- \frac{4 x y}{( x ^{2} + y ^{2} ) ^{3}}

Solution steps

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

Treat

x

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

- \frac{2}{( x ^{2} + y ^{2} ) ^{3}} \frac{\partial}{\partial y} (x^{2} + y^{2})

= - \frac{2}{( x ^{2} + y ^{2} ) ^{3}} \frac{\partial}{\partial y} (x^{2} + y^{2})

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

= x (- \frac{2}{( x ^{2} + y ^{2} ) ^{3}} \cdot 2 y)

Simplify

x (- \frac{2}{( x ^{2} + y ^{2} ) ^{3}} \cdot 2 y) : - \frac{4 x y}{( x ^{2} + y ^{2} ) ^{3}}

= - \frac{4 x y}{( x ^{2} + y ^{2} ) ^{3}}

What is (\partial)/(\partial y)(x/((x^2+y^2)^2)) ?
The answer to (\partial)/(\partial y)(x/((x^2+y^2)^2)) is -(4xy)/((x^2+y^2)^3)