What is (\partial)/(\partial x)(x*e^{xy}) ?

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

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(\partial)/(\partial x)(x*e^{xy})

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

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

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} y

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

Simplify

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

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