What is (\partial)/(\partial z)(\sqrt[3]{xyz}) ?

The answer to (\partial)/(\partial z)(\sqrt[3]{xyz}) is (x^{1/3}y^{1/3})/(3z^{2/3)}

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(\partial)/(\partial z)(\sqrt[3]{xyz})

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

Solution steps

Treat

x, y

as constants

Apply the chain rule:

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

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

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

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

Simplify

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

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

What is (\partial)/(\partial z)(\sqrt[3]{xyz}) ?
The answer to (\partial)/(\partial z)(\sqrt[3]{xyz}) is (x^{1/3}y^{1/3})/(3z^{2/3)}