in constrained Lagrangian optimization what is a general way to figure out how the optimal point varies with respect to parameters of the constraints? For example maximize x*y when x + y < k, and I wanted to find dx*/dk and dy*/dk where (x*,y*) is the optimal point. How would I do that?
The optimal point may not be differentiable at a value of $k$ where a constraint goes from binding to non-binding. At any other value of $k$ just differentiate with respect to $k$ the (equality) first order conditions for the constraints which bind and solve for $\partial x^*/\partial k$ etc.
Take a look at any standard optimization book, for eg. Luenberger Linear and Non-linear Programming or Sundaram's A First Course in Optimization Theory,