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

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

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

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

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

Solution steps

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

Treat

x, y

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 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}}

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

Simplify

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

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

cos (x), 2 - cos (x), [0, 2 π]

- 16 t^{2} + 44 t + 31

x \to 2 lim (3 x - 5)

x 1 - y^{2} d x = d y

\frac{d y}{d x} = y x sin (x^{2})

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