What is (\partial)/(\partial y)(5x^{y/z}) ?

The answer to (\partial)/(\partial y)(5x^{y/z}) is (5x^{y/z}ln(x))/z

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(\partial)/(\partial y)(5x^{y/z})

\frac{\partial}{\partial y} (5 x^{\frac{y}{z}})

\frac{5 x ^{\frac{y}{z}} ln ( x )}{z}

Solution steps

\frac{\partial}{\partial y} (5 x^{\frac{y}{z}})

Treat

x, z

as constants

Take the constant out:

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

= 5 \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)}

= 5 \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}

= 5 e^{\frac{y}{z} l n (x)} \frac{ln ( x )}{z}

Simplify

5 e^{\frac{y}{z} l n (x)} \frac{ln ( x )}{z} : \frac{5 x ^{\frac{y}{z}} ln ( x )}{z}

= \frac{5 x ^{\frac{y}{z}} ln ( x )}{z}

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

\frac{d}{d x} (\frac{x ^{2} + 1}{x - 3})

x \to - 8 lim (- \frac{3}{7} x + \frac{10}{11})

\frac{d}{d x} (\frac{7}{x - 1})

What is (\partial)/(\partial y)(5x^{y/z}) ?
The answer to (\partial)/(\partial y)(5x^{y/z}) is (5x^{y/z}ln(x))/z