What is (\partial)/(\partial y)(1/(x^2+y^2)) ?

The answer to (\partial)/(\partial y)(1/(x^2+y^2)) is -(2y)/((x^2+y^2)^2)

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(\partial)/(\partial y)(1/(x^2+y^2))

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

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

Solution steps

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

Treat

x

as a constant

Apply exponent rule:

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

= \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

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

Simplify

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

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

n \to \infty lim (an)

x \to 3 lim (\frac{7}{x - 3})

\int - \frac{2}{x} d x

\frac{d y}{d t} = \frac{2}{t} y + 3 t^{2} cos (3 t)

\int_{1}^{\infty} x^{- \frac{2}{3}} d x

What is (\partial)/(\partial y)(1/(x^2+y^2)) ?
The answer to (\partial)/(\partial y)(1/(x^2+y^2)) is -(2y)/((x^2+y^2)^2)