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 \frac{1}{cos ^{2} ( b ) x} d x

x \to 2 lim (- 5)

\int - cos (2 t) d t

\int \frac{1}{( 1 - x ) ( 2 - x )} d x

\frac{1}{cos ^{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