What is (\partial)/(\partial y)(3x^{y/z}) ?

The answer to (\partial)/(\partial y)(3x^{y/z}) is (3x^{y/z}ln(x))/z

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(\partial)/(\partial y)(3x^{y/z})

\frac{\partial}{\partial y} (3 x^{\frac{y}{z}})

\frac{3 x ^{\frac{y}{z}} ln ( x )}{z}

Solution steps

\frac{\partial}{\partial y} (3 x^{\frac{y}{z}})

Treat

x, z

as constants

Take the constant out:

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

= 3 \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)}

= 3 \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}

= 3 e^{\frac{y}{z} l n (x)} \frac{ln ( x )}{z}

Simplify

3 e^{\frac{y}{z} l n (x)} \frac{ln ( x )}{z} : \frac{3 x ^{\frac{y}{z}} ln ( x )}{z}

= \frac{3 x ^{\frac{y}{z}} ln ( x )}{z}

\frac{\partial}{\partial y} (2 e^{- 2 x y} + x y^{3})

x, 20 - x^{2}

x \to 2 lim (2 x - x^{2})

2 x y^{^{'}} - 3 y = 0

\frac{d y}{d x} = 3 x^{2} y^{4}

What is (\partial)/(\partial y)(3x^{y/z}) ?
The answer to (\partial)/(\partial y)(3x^{y/z}) is (3x^{y/z}ln(x))/z