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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

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