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

The answer to (\partial)/(\partial y)((x^2+y^2)^{3/2}) is 3y(x^2+y^2)^{1/2}

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

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What is (\partial)/(\partial y)((x^2+y^2)^{3/2}) ?
The answer to (\partial)/(\partial y)((x^2+y^2)^{3/2}) is 3y(x^2+y^2)^{1/2}