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}

f (x) = (\frac{x}{2})^{4}

f (x) = (x - 1)^{\frac{2}{3}}

n = 0 \sum \infty x^{4 n}

\frac{d}{d x} (3 y x y)

\int 5 cos^{3} (3 x) d x

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