What is (\partial)/(\partial z)((xy^2)/(z^3)) ?

The answer to (\partial)/(\partial z)((xy^2)/(z^3)) is -(3xy^2)/(z^4)

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(\partial)/(\partial z)((xy^2)/(z^3))

\frac{\partial}{\partial z} (\frac{x y ^{2}}{z ^{3}})

- \frac{3 x y ^{2}}{z ^{4}}

Solution steps

\frac{\partial}{\partial z} (\frac{x y ^{2}}{z ^{3}})

Treat

x, y

as constants

Take the constant out:

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

= x y^{2} \frac{\partial}{\partial z} (\frac{1}{z ^{3}})

Apply exponent rule:

\frac{1}{a} = a^{- 1}

= x y^{2} \frac{\partial}{\partial z} (z^{- 3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= x y^{2} (- 3 z^{- 3 - 1})

Simplify

x y^{2} (- 3 z^{- 3 - 1}) : - \frac{3 x y ^{2}}{z ^{4}}

= - \frac{3 x y ^{2}}{z ^{4}}

What is (\partial)/(\partial z)((xy^2)/(z^3)) ?
The answer to (\partial)/(\partial z)((xy^2)/(z^3)) is -(3xy^2)/(z^4)