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

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

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

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

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

Solution steps

\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})^{- \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

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

Simplify

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

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

\int_{1200}^{1210} 0.1 x d x

x \to 0 lim (x^{3} \cdot ln (x))

\frac{\partial}{\partial z} (3)

sin (x^{2})

f (s) = 8^{2 s - 1}

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