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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

2 y (- \frac{1}{( x ^{2} + y ^{2} ) ^{2}} \cdot 2 x) : - \frac{4 y x}{( x ^{2} + y ^{2} ) ^{2}}

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

x \to 0 + lim (cos (x) - 1)

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