What is (\partial)/(\partial z)(x^2ye^{y/z}) ?

The answer to (\partial)/(\partial z)(x^2ye^{y/z}) is -(e^{y/z}x^2y^2)/(z^2)

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(\partial)/(\partial z)(x^2ye^{y/z})

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

- \frac{e ^{\frac{y}{z}} x ^{2} y ^{2}}{z ^{2}}

Solution steps

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

Treat

x, y

as constants

Take the constant out:

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

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

Apply the chain rule:

e^{\frac{y}{z}} \frac{\partial}{\partial z} (\frac{y}{z})

= e^{\frac{y}{z}} \frac{\partial}{\partial z} (\frac{y}{z})

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

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

Simplify

x^{2} y e^{\frac{y}{z}} (- \frac{y}{z ^{2}}) : - \frac{e ^{\frac{y}{z}} x ^{2} y ^{2}}{z ^{2}}

= - \frac{e ^{\frac{y}{z}} x ^{2} y ^{2}}{z ^{2}}

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What is (\partial)/(\partial z)(x^2ye^{y/z}) ?
The answer to (\partial)/(\partial z)(x^2ye^{y/z}) is -(e^{y/z}x^2y^2)/(z^2)