What is (\partial)/(\partial y)(7x^{y/z}) ?

The answer to (\partial)/(\partial y)(7x^{y/z}) is (7x^{y/z}ln(x))/z

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(\partial)/(\partial y)(7x^{y/z})

\frac{\partial}{\partial y} (7 x^{\frac{y}{z}})

\frac{7 x ^{\frac{y}{z}} ln ( x )}{z}

Solution steps

\frac{\partial}{\partial y} (7 x^{\frac{y}{z}})

Treat

x, z

as constants

Take the constant out:

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

= 7 \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)}

= 7 \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}

= 7 e^{\frac{y}{z} l n (x)} \frac{ln ( x )}{z}

Simplify

7 e^{\frac{y}{z} l n (x)} \frac{ln ( x )}{z} : \frac{7 x ^{\frac{y}{z}} ln ( x )}{z}

= \frac{7 x ^{\frac{y}{z}} ln ( x )}{z}

y^{^{'}} + \frac{2 y}{x} = \frac{2}{x}

\int \frac{u}{u + 1} d u

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

\int x 4 x + 1 d x

f (x) = 5 x + 7 x

What is (\partial)/(\partial y)(7x^{y/z}) ?
The answer to (\partial)/(\partial y)(7x^{y/z}) is (7x^{y/z}ln(x))/z