What is (\partial)/(\partial x)(xe^{2x+3y}) ?

The answer to (\partial)/(\partial x)(xe^{2x+3y}) is e^{2x+3y}+2e^{2x+3y}x

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(\partial)/(\partial x)(xe^{2x+3y})

\frac{\partial}{\partial x} (x e^{2 x + 3 y})

e^{2 x + 3 y} + 2 e^{2 x + 3 y} x

Solution steps

\frac{\partial}{\partial x} (x e^{2 x + 3 y})

Treat

y

as a constant

Apply the Product Rule:

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

f = x, g = e^{2 x + 3 y}

= \frac{\partial}{\partial x} (x) e^{2 x + 3 y} + \frac{\partial}{\partial x} (e^{2 x + 3 y}) x

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

\frac{\partial}{\partial x} (e^{2 x + 3 y}) = e^{2 x + 3 y} \cdot 2

= 1 \cdot e^{2 x + 3 y} + e^{2 x + 3 y} \cdot 2 x

Simplify

= e^{2 x + 3 y} + 2 e^{2 x + 3 y} x

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What is (\partial)/(\partial x)(xe^{2x+3y}) ?
The answer to (\partial)/(\partial x)(xe^{2x+3y}) is e^{2x+3y}+2e^{2x+3y}x