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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = e^{\frac{x}{y}}

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

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

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

= 1 \cdot e^{\frac{x}{y}} + \frac{e ^{\frac{x}{y}}}{y} x

Simplify

1 \cdot e^{\frac{x}{y}} + \frac{e ^{\frac{x}{y}}}{y} x : e^{\frac{x}{y}} + \frac{e ^{\frac{x}{y}} x}{y}

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

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