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

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

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(\partial)/(\partial x)(x*e^{y/x})

\frac{\partial}{\partial x} (x \cdot 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}

f^{^{'}} (x) = \frac{4}{1 - 2 x ^{2}}

sin (t^{2}) ln (t^{3})

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

\int 4 e^{x} d x

x + y^{2} = 30, x + y = 0

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