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

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

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

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

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

Solution steps

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

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

Simplify

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

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

x \to 0 lim (x ln (5 x))

\int \frac{x}{2 + x ^{4}} d x

\frac{d}{d y} ((\frac{1}{y ^{2}} - \frac{7}{y ^{4}}) (y + 5 y^{3}))

\frac{1}{( s ^{2} - 2 s )}

\int_{0}^{5} \frac{1}{x ^{2} - 6 x + 5} d x

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