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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

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

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

Simplify

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

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

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