What is (\partial)/(\partial z)(6x^{y/z}) ?

The answer to (\partial)/(\partial z)(6x^{y/z}) is -(6yx^{y/z}ln(x))/(z^2)

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(\partial)/(\partial z)(6x^{y/z})

\frac{\partial}{\partial z} (6 x^{\frac{y}{z}})

- \frac{6 y x ^{\frac{y}{z}} ln ( x )}{z ^{2}}

Solution steps

\frac{\partial}{\partial z} (6 x^{\frac{y}{z}})

Treat

x, y

as constants

Take the constant out:

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

= 6 \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)}

= 6 \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}}

= 6 e^{\frac{y}{z} l n (x)} (- \frac{y ln ( x )}{z ^{2}})

Simplify

6 e^{\frac{y}{z} l n (x)} (- \frac{y ln ( x )}{z ^{2}}) : - \frac{6 y x ^{\frac{y}{z}} ln ( x )}{z ^{2}}

= - \frac{6 y x ^{\frac{y}{z}} ln ( x )}{z ^{2}}

\frac{1}{s + 3} - \frac{1}{s + 4}

\frac{\partial}{\partial s} (\frac{( 2 r + s )}{t})

\frac{d}{d y} (x sin (x y))

x \to 1 lim (\frac{1}{x - 2})

\frac{\partial}{\partial x} (3 y sin (7 y^{2} x))

What is (\partial)/(\partial z)(6x^{y/z}) ?
The answer to (\partial)/(\partial z)(6x^{y/z}) is -(6yx^{y/z}ln(x))/(z^2)