What is (\partial)/(\partial y)((1+xy)e^{xy}) ?

The answer to (\partial)/(\partial y)((1+xy)e^{xy}) is 2e^{xy}x+e^{xy}x^2y

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(\partial)/(\partial y)((1+xy)e^{xy})

\frac{\partial}{\partial y} ((1 + x y) e^{x y})

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

Solution steps

\frac{\partial}{\partial y} ((1 + x y) e^{x y})

Treat

x

as a constant

Apply the Product Rule:

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

f = 1 + x y, g = e^{x y}

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

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

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

= x e^{x y} + e^{x y} x (1 + x y)

Simplify

x e^{x y} + e^{x y} x (1 + x y) : 2 e^{x y} x + e^{x y} x^{2} y

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

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What is (\partial)/(\partial y)((1+xy)e^{xy}) ?
The answer to (\partial)/(\partial y)((1+xy)e^{xy}) is 2e^{xy}x+e^{xy}x^2y