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

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

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

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

e^{x + 2 y} + e^{x + 2 y} (x + y)

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

= e^{x + 2 y} + e^{x + 2 y} (x + y)

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