To find $\frac{dy}{dx}$ using partial derivatives for the given equation:

$e^{xy} + y e^{y} = 6$

we will implicitly differentiate both sides of the equation with respect to $x$, treating $y$ as a function of $x$ (i.e., $y = y(x)$).

Step 1: Differentiate the first term $e^{xy}$

Using the chain rule: $\frac{d}{dx} \left( e^{xy} \right) = e^{xy} \cdot \frac{d}{dx}(xy) = e^{xy} \left( y + x \frac{dy}{dx} \right)$ (where $\frac{dy}{dx}$ is the derivative of $y$ with respect to $x$).

Step 2: Differentiate the second term $y e^y$

This is a product of two functions, so we apply