What is (\partial)/(\partial y)((xy)^{1/3}) ?

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

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(\partial)/(\partial y)((xy)^{1/3})

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

\frac{x ^{\frac{1}{3}}}{3 y ^{\frac{2}{3}}}

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

= \frac{x ^{\frac{1}{3}}}{3 y ^{\frac{2}{3}}}

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