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

The answer to (\partial)/(\partial z)(4x^{y/z}) is -(4yx^{y/z}ln(x))/(z^2)

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

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

- \frac{4 y x ^{\frac{y}{z}} ln ( x )}{z ^{2}}

Solution steps

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

Treat

x, y

as constants

Take the constant out:

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

= 4 \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)}

= 4 \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}}

= 4 e^{\frac{y}{z} l n (x)} (- \frac{y ln ( x )}{z ^{2}})

Simplify

4 e^{\frac{y}{z} l n (x)} (- \frac{y ln ( x )}{z ^{2}}) : - \frac{4 y x ^{\frac{y}{z}} ln ( x )}{z ^{2}}

= - \frac{4 y x ^{\frac{y}{z}} ln ( x )}{z ^{2}}

2 x - 10, [4, 8]

cos (\frac{3}{2} x)

2000 (1 - \frac{x}{600})^{2}

\frac{d}{d x} (ln (x^{2} + 19))

\int \frac{3}{6 x - x ^{2}} d x

What is (\partial)/(\partial z)(4x^{y/z}) ?
The answer to (\partial)/(\partial z)(4x^{y/z}) is -(4yx^{y/z}ln(x))/(z^2)