What is (\partial)/(\partial x)((y/z)^{1/x}) ?

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

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(\partial)/(\partial x)((y/z)^{1/x})

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

- \frac{y ^{\frac{1}{x}} ln ( \frac{y}{z} )}{x ^{2} z ^{\frac{1}{x}}}

Solution steps

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

Treat

y, z

as constants

Apply exponent rule:

a^{b} = e^{b l n (a)}

(\frac{y}{z})^{\frac{1}{x}} = e^{\frac{1}{x} l n (\frac{y}{z})}

= \frac{\partial}{\partial x} (e^{\frac{1}{x} l n (\frac{y}{z})})

Apply the chain rule:

e^{\frac{1}{x} l n (\frac{y}{z})} \frac{\partial}{\partial x} (\frac{1}{x} ln (\frac{y}{z}))

= e^{\frac{1}{x} l n (\frac{y}{z})} \frac{\partial}{\partial x} (\frac{1}{x} ln (\frac{y}{z}))

\frac{\partial}{\partial x} (\frac{1}{x} ln (\frac{y}{z})) = - \frac{ln ( \frac{y}{z} )}{x ^{2}}

= e^{\frac{1}{x} l n (\frac{y}{z})} (- \frac{ln ( \frac{y}{z} )}{x ^{2}})

Simplify

e^{\frac{1}{x} l n (\frac{y}{z})} (- \frac{ln ( \frac{y}{z} )}{x ^{2}}) : - \frac{y ^{\frac{1}{x}} ln ( \frac{y}{z} )}{x ^{2} z ^{\frac{1}{x}}}

= - \frac{y ^{\frac{1}{x}} ln ( \frac{y}{z} )}{x ^{2} z ^{\frac{1}{x}}}

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