What is (\partial)/(\partial y)((3x+3y)e^y) ?

The answer to (\partial)/(\partial y)((3x+3y)e^y) is 3e^y+e^y(3x+3y)

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(\partial)/(\partial y)((3x+3y)e^y)

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

3 e^{y} + e^{y} (3 x + 3 y)

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

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

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

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

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

= 3 e^{y} + e^{y} (3 x + 3 y)

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