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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

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