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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

\frac{\partial}{\partial x} (x^{2}) = 2 x

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

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

Simplify

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

= 2 e^{\frac{y}{x}} x - e^{\frac{y}{x}} y

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