What is (\partial)/(\partial x)((xyz)^{1/2}) ?

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

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(\partial)/(\partial x)((xyz)^{1/2})

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

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

Solution steps

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

Treat

y, z

as constants

Apply the chain rule:

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

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

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

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

Simplify

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

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

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