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

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

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

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

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

Solution steps

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

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

Simplify

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

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

\int \frac{1}{x ^{- 0.1}} d x

x \to \infty lim (x \cdot (arctan (e^{x}) - \frac{π}{2}))

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

\int \frac{1}{( x ^{4} - 1 ) ^{2}} d x

y = x + x + x

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