We are tasked with finding the partial derivatives of the function $P = u^2 + v^2 + w^2$, where $u = x e^y$, $v = y e^x$, and $w = e^{xy}$, using the Chain Rule.

Step 1: Compute the partial derivative of $P$ with respect to $x$

The Chain Rule tells us that:

$\frac{\partial P}{\partial x} = 2u \frac{\partial u}{\partial x} + 2v \frac{\partial v}{\partial x} + 2w \frac{\partial w}{\partial x}$

1.1: Derivatives of $u$, $v$, and $w$ with respect to $x$

• $u = x e^y$