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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

\frac{\partial}{\partial y} (x) = 0

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

= 0 - \frac{2 x y}{( x ^{2} + y ^{2} ) ^{2}}

Simplify

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

\frac{d y}{d x} = 4 e^{- 3 x}

\int e^{x} \cdot sin (e^{x}) d x

x \to 3 - lim (- (3)^{2} + 5 (3) - 5)

8 x - 2

\int_{0}^{1} (3 e^{- x} - e^{- 2 x}) d x

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