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} )})

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}}

\int (x^{2}) d x

\frac{d}{d x} ((2 x)^{2 x})

\frac{d}{d x} (4 x^{3} sec (7 x))

t an g e n t x^{2} + 16, at x = 3

\int \frac{1}{3} x^{3} 9 - x^{2} d 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)