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

The answer to (\partial)/(\partial y)(y/((x^2+y^2)^{3/2)}) is (-2y^2+x^2)/((x^2+y^2)^{5/2)}

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

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

\frac{- 2 y ^{2} + x ^{2}}{( x ^{2} + y ^{2} ) ^{\frac{5}{2}}}

Solution steps

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

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

\frac{\partial}{\partial y} ((x^{2} + y^{2})^{\frac{3}{2}}) = 3 y (x^{2} + y^{2})^{\frac{1}{2}}

= \frac{1 \cdot ( x ^{2} + y ^{2} ) ^{\frac{3}{2}} - 3 y ( x ^{2} + y ^{2} ) ^{\frac{1}{2}} y}{( ( x ^{2} + y ^{2} ) ^{\frac{3}{2}} ) ^{2}}

Simplify

\frac{1 \cdot ( x ^{2} + y ^{2} ) ^{\frac{3}{2}} - 3 y ( x ^{2} + y ^{2} ) ^{\frac{1}{2}} y}{( ( x ^{2} + y ^{2} ) ^{\frac{3}{2}} ) ^{2}} : \frac{- 2 y ^{2} + x ^{2}}{( x ^{2} + y ^{2} ) ^{\frac{5}{2}}}

= \frac{- 2 y ^{2} + x ^{2}}{( x ^{2} + y ^{2} ) ^{\frac{5}{2}}}

What is (\partial)/(\partial y)(y/((x^2+y^2)^{3/2)}) ?
The answer to (\partial)/(\partial y)(y/((x^2+y^2)^{3/2)}) is (-2y^2+x^2)/((x^2+y^2)^{5/2)}