What is (\partial)/(\partial x)((3x+2y)e^{x/y}) ?

The answer to (\partial)/(\partial x)((3x+2y)e^{x/y}) is (3e^{x/y}x+5e^{x/y}y)/y

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(\partial)/(\partial x)((3x+2y)e^{x/y})

\frac{\partial}{\partial x} ((3 x + 2 y) e^{\frac{x}{y}})

\frac{3 e ^{\frac{x}{y}} x + 5 e ^{\frac{x}{y}} y}{y}

Solution steps

\frac{\partial}{\partial x} ((3 x + 2 y) e^{\frac{x}{y}})

Treat

y

as a constant

Apply the Product Rule:

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

f = 3 x + 2 y, g = e^{\frac{x}{y}}

= \frac{\partial}{\partial x} (3 x + 2 y) e^{\frac{x}{y}} + \frac{\partial}{\partial x} (e^{\frac{x}{y}}) (3 x + 2 y)

\frac{\partial}{\partial x} (3 x + 2 y) = 3

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

= 3 e^{\frac{x}{y}} + \frac{e ^{\frac{x}{y}}}{y} (3 x + 2 y)

Simplify

3 e^{\frac{x}{y}} + \frac{e ^{\frac{x}{y}}}{y} (3 x + 2 y) : \frac{3 e ^{\frac{x}{y}} x + 5 e ^{\frac{x}{y}} y}{y}

= \frac{3 e ^{\frac{x}{y}} x + 5 e ^{\frac{x}{y}} y}{y}

What is (\partial)/(\partial x)((3x+2y)e^{x/y}) ?
The answer to (\partial)/(\partial x)((3x+2y)e^{x/y}) is (3e^{x/y}x+5e^{x/y}y)/y