What is (\partial)/(\partial y)(9x^{y/z}) ?

The answer to (\partial)/(\partial y)(9x^{y/z}) is (9x^{y/z}ln(x))/z

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(\partial)/(\partial y)(9x^{y/z})

\frac{\partial}{\partial y} (9 x^{\frac{y}{z}})

\frac{9 x ^{\frac{y}{z}} ln ( x )}{z}

Solution steps

\frac{\partial}{\partial y} (9 x^{\frac{y}{z}})

Treat

x, z

as constants

Take the constant out:

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

= 9 \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)}

= 9 \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}

= 9 e^{\frac{y}{z} l n (x)} \frac{ln ( x )}{z}

Simplify

9 e^{\frac{y}{z} l n (x)} \frac{ln ( x )}{z} : \frac{9 x ^{\frac{y}{z}} ln ( x )}{z}

= \frac{9 x ^{\frac{y}{z}} ln ( x )}{z}

\frac{d}{d x} (\frac{sin ( 3 x )}{cos ( 2 x )})

arcsin (\frac{x}{9})

f (x) = \frac{e ^{x}}{x}, e, at x = 1

\int (9 x - 6)^{2} d x

- x e^{x}

What is (\partial)/(\partial y)(9x^{y/z}) ?
The answer to (\partial)/(\partial y)(9x^{y/z}) is (9x^{y/z}ln(x))/z