What is (\partial)/(\partial z)(5x^{y/z}) ?

The answer to (\partial)/(\partial z)(5x^{y/z}) is -(5yx^{y/z}ln(x))/(z^2)

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(\partial)/(\partial z)(5x^{y/z})

\frac{\partial}{\partial z} (5 x^{\frac{y}{z}})

- \frac{5 y x ^{\frac{y}{z}} ln ( x )}{z ^{2}}

Solution steps

\frac{\partial}{\partial z} (5 x^{\frac{y}{z}})

Treat

x, y

as constants

Take the constant out:

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

= 5 \frac{\partial}{\partial z} (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)}

= 5 \frac{\partial}{\partial z} (e^{\frac{y}{z} l n (x)})

Apply the chain rule:

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

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

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

= 5 e^{\frac{y}{z} l n (x)} (- \frac{y ln ( x )}{z ^{2}})

Simplify

5 e^{\frac{y}{z} l n (x)} (- \frac{y ln ( x )}{z ^{2}}) : - \frac{5 y x ^{\frac{y}{z}} ln ( x )}{z ^{2}}

= - \frac{5 y x ^{\frac{y}{z}} ln ( x )}{z ^{2}}

What is (\partial)/(\partial z)(5x^{y/z}) ?
The answer to (\partial)/(\partial z)(5x^{y/z}) is -(5yx^{y/z}ln(x))/(z^2)