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

The answer to (\partial)/(\partial y)(1/(sqrt(x^2+y^2))) is -y/((x^2+y^2)^{3/2)}

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

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

- \frac{y}{( x ^{2} + y ^{2} ) ^{\frac{3}{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})^{- \frac{1}{2}})

Apply the chain rule:

- \frac{1}{2 ( x ^{2} + y ^{2} ) ^{\frac{3}{2}}} \frac{\partial}{\partial y} (x^{2} + y^{2})

= - \frac{1}{2 ( x ^{2} + y ^{2} ) ^{\frac{3}{2}}} \frac{\partial}{\partial y} (x^{2} + y^{2})

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

= - \frac{1}{2 ( x ^{2} + y ^{2} ) ^{\frac{3}{2}}} \cdot 2 y

Simplify

- \frac{1}{2 ( x ^{2} + y ^{2} ) ^{\frac{3}{2}}} \cdot 2 y : - \frac{y}{( x ^{2} + y ^{2} ) ^{\frac{3}{2}}}

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

x \to 0 lim (\frac{1}{x} + \frac{1}{x ^{2} - x})

x \to \infty lim (\frac{3}{4})

x \to - 2 lim (12 x + 1)

\frac{d y}{d x} + \frac{y}{100 + x} = 4

\int - \frac{3 x - 1}{( x - 5 ) ( x - 7 ) ^{2}} d x

What is (\partial)/(\partial y)(1/(sqrt(x^2+y^2))) ?
The answer to (\partial)/(\partial y)(1/(sqrt(x^2+y^2))) is -y/((x^2+y^2)^{3/2)}