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

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

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

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

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

Solution steps

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

Simplify

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

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

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