What is (\partial)/(\partial y)(4x^{y/z}) ?

The answer to (\partial)/(\partial y)(4x^{y/z}) is (4x^{y/z}ln(x))/z

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(\partial)/(\partial y)(4x^{y/z})

\frac{\partial}{\partial y} (4 x^{\frac{y}{z}})

\frac{4 x ^{\frac{y}{z}} ln ( x )}{z}

Solution steps

\frac{\partial}{\partial y} (4 x^{\frac{y}{z}})

Treat

x, z

as constants

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 4 \frac{\partial}{\partial y} (x^{\frac{y}{z}})

Apply exponent rule:

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

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

= 4 \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}

= 4 e^{\frac{y}{z} l n (x)} \frac{ln ( x )}{z}

Simplify

4 e^{\frac{y}{z} l n (x)} \frac{ln ( x )}{z} : \frac{4 x ^{\frac{y}{z}} ln ( x )}{z}

= \frac{4 x ^{\frac{y}{z}} ln ( x )}{z}

What is (\partial)/(\partial y)(4x^{y/z}) ?
The answer to (\partial)/(\partial y)(4x^{y/z}) is (4x^{y/z}ln(x))/z