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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the Quotient Rule:

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

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

\frac{\partial}{\partial y} (y) = 1

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

= \frac{1 \cdot ( x ^{2} + y ^{2} ) - 2 yy}{( x ^{2} + y ^{2} ) ^{2}}

1 \cdot (x^{2} + y^{2}) - 2 yy = x^{2} - y^{2}

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

f (x) = 4 x^{6} - 7 x^{5}

5 x sin (π x)

\int \frac{x ^{3}}{x ^{2} + 100} d x

t \to - 2 lim (\frac{t ^{3} + 8}{t + 2})

\frac{d}{d x} (sin (arccos (3 x)))

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