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

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

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

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

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

Solution steps

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

Treat

x, z

as constants

Take the constant out:

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

= 6 \frac{\partial}{\partial y} (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 y} (e^{\frac{y}{z} l n (x)})

Apply the chain rule:

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

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

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

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

Simplify

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

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

\int \frac{2}{\frac{1}{2} + x ^{2}} d x

\int \frac{x ^{4} + x ^{2} + 4}{x ^{2} - 4} d x

\int \frac{4 x ^{2}}{x ^{3} + 16} d x

y = x^{\frac{5}{3}} - x^{\frac{2}{3}}

x \to - 4 lim (x (x + 3) (x + 2))

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