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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Product Rule:

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

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

= 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}

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

Simplify

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

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

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