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

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

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

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

\frac{y ^{\frac{x}{z}} ln ( y )}{z}

Solution steps

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

Treat

y, z

as constants

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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