What is (\partial)/(\partial z)(9x^{y/z}) ?

The answer to (\partial)/(\partial z)(9x^{y/z}) is -(9yx^{y/z}ln(x))/(z^2)

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(\partial)/(\partial z)(9x^{y/z})

\frac{\partial}{\partial z} (9 x^{\frac{y}{z}})

- \frac{9 y x ^{\frac{y}{z}} ln ( x )}{z ^{2}}

Solution steps

\frac{\partial}{\partial z} (9 x^{\frac{y}{z}})

Treat

x, y

as constants

Take the constant out:

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

= 9 \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)}

= 9 \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}}

= 9 e^{\frac{y}{z} l n (x)} (- \frac{y ln ( x )}{z ^{2}})

Simplify

9 e^{\frac{y}{z} l n (x)} (- \frac{y ln ( x )}{z ^{2}}) : - \frac{9 y x ^{\frac{y}{z}} ln ( x )}{z ^{2}}

= - \frac{9 y x ^{\frac{y}{z}} ln ( x )}{z ^{2}}

\int x^{2} \cdot e^{x^{2}} d x

x \to 2 lim (3^{x} + ln (x - 1))

f (2) = \frac{4}{1 + x ^{2}}, at x = 1

t y^{^{'}} + 2 y = sin (t)

\int 6 x (2 x^{2} + 9)^{\frac{1}{2}} d x

What is (\partial)/(\partial z)(9x^{y/z}) ?
The answer to (\partial)/(\partial z)(9x^{y/z}) is -(9yx^{y/z}ln(x))/(z^2)