What is (\partial)/(\partial y)(xe^{-y/x}) ?

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

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(\partial)/(\partial y)(xe^{-y/x})

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the chain rule:

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

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

\frac{\partial}{\partial y} (- \frac{y}{x}) = - \frac{1}{x}

= x e^{- \frac{y}{x}} (- \frac{1}{x})

Simplify

x e^{- \frac{y}{x}} (- \frac{1}{x}) : - e^{- \frac{y}{x}}

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

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