I need to compute the derivative matrix for
$\displaystyle f(x,y) = (xe^y + cos(y), x , x+e^y)$
would this be $\displaystyle f : \mathbb{R}^2 \to \mathbb{R}^3$ making a 3x2 matrix ??
Yes
$\displaystyle \begin{pmatrix}\tfrac{\partial (xe^y + \cos y)}{\partial x} & \tfrac{\partial (xe^y + \cos y)}{\partial y} \\ \tfrac{\partial (x)}{\partial x} & \tfrac{\partial (x)}{\partial y} \\ \tfrac{\partial(x+e^y)}{\partial x} & \tfrac{\partial(x+e^y)}{\partial y}\end{pmatrix}$
Note: By "derivative matrix", I assume you are looking for the Jacobian. There could be other derivative matrices (in theory).
This is exactly the same as in your thread
linear approximation of mapping