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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

= 1 \cdot e^{\frac{y}{x}} + (- \frac{y e ^{\frac{y}{x}}}{x ^{2}}) x

Simplify

1 \cdot e^{\frac{y}{x}} + (- \frac{y e ^{\frac{y}{x}}}{x ^{2}}) x : e^{\frac{y}{x}} - \frac{e ^{\frac{y}{x}} y}{x}

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

\int (x \cdot e^{2 \cdot x}) d x

\frac{\partial}{\partial y} (y - ln (x^{2} + y^{2}))

y = \frac{3}{x ^{2} - 1}

x \to 0 + lim (6 (cot (x) - \frac{1}{x}))

\frac{d}{d x} (- e^{- 2 x})

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