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

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

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

e^{x + y} + e^{x + y} x

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

f = x, g = e^{x + y}

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

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

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

= 1 \cdot e^{x + y} + e^{x + y} x

Simplify

= e^{x + y} + e^{x + y} x

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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