What is (\partial)/(\partial x)((x^3)/(x^2+y^2)) ?

The answer to (\partial)/(\partial x)((x^3)/(x^2+y^2)) is (x^4+3x^2y^2)/((x^2+y^2)^2)

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(\partial)/(\partial x)((x^3)/(x^2+y^2))

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

\frac{x ^{4} + 3 x ^{2} y ^{2}}{( x ^{2} + y ^{2} ) ^{2}}

Solution steps

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

Treat

y

as a constant

Apply the Quotient Rule:

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

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

\frac{\partial}{\partial x} (x^{3}) = 3 x^{2}

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

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

Simplify

\frac{3 x ^{2} ( x ^{2} + y ^{2} ) - 2 x x ^{3}}{( x ^{2} + y ^{2} ) ^{2}} : \frac{x ^{4} + 3 x ^{2} y ^{2}}{( x ^{2} + y ^{2} ) ^{2}}

= \frac{x ^{4} + 3 x ^{2} y ^{2}}{( x ^{2} + y ^{2} ) ^{2}}

\int (x + 2)^{11} d x

y^{^{''}} + 2 y^{^{'}} + 5 y = 9 sin (2 t)

\int ln ∣ x + 1 ∣ d x

\frac{d}{d x} ((x^{2} - 3 x)^{5})

(12 x^{2} + y - 1) d x - (6 y - x) d y = 0, y (1) = 0

What is (\partial)/(\partial x)((x^3)/(x^2+y^2)) ?
The answer to (\partial)/(\partial x)((x^3)/(x^2+y^2)) is (x^4+3x^2y^2)/((x^2+y^2)^2)