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

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

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

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

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

Solution steps

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

Simplify

\frac{- x}{x ^{2} + y ^{2}} : - \frac{x}{x ^{2} + y ^{2}}

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

sin (x y) d x = 2 x d y + 2 y d x

\int \frac{x ^{2}}{x + 2} d x

\frac{d}{d x} (\frac{( x ^{2} + 2 x )}{x + 1})^{5}

\frac{\partial}{\partial x} (y arcsin (\frac{x}{y}))

x \to - \infty lim (\frac{4 x ^{4} - 3 + 6 x}{1 - 2 x})

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