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}}}

\int 2500 - x^{2} d x

\frac{d}{d x} ((2 x^{2} - 1)^{3} \cdot (1 - x^{2})^{- 3})

\frac{d}{d x} (2 + x^{2})

\frac{d}{d x} (\frac{( x ^{3} + 2 x ) ^{3}}{2 x - 4})

\frac{1}{cos ( x )}

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)}