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

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

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

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

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

Solution steps

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

Treat

x, z

as constants

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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

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

\frac{d}{d x} (- x - \frac{16}{x})

f (x) = sin (ln (x))

\frac{d P}{d t} = P \cdot (0.0005 \cdot P - 0.1), P (0) = 300

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