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}}

\int_{1}^{\infty} \frac{5}{x ^{3}} d x

\int \frac{sec ( x ) tan ( x )}{1 + sec ( x )} d x

\frac{d}{d x} (\frac{1}{2 ^{x}})

\int_{0}^{\frac{π}{27}} x tan^{2} (9 x) d x

x \to 0 lim (\frac{tan ( 10 x )}{x})

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