Gradients for nonlinear optimization constraints

Hello,

I'm currently getting into gradients for constraints of nonlinear optimization and I have some struggle to understand it.

I have the Problem, that my optimization tends to turn to not feasible results. I want to solve that with gradients for my constraints, which are pretty strict. My optimization contains 10 variables, I have one inequality and two equality constraints. None of the constraints is limiting all of the variables directly.

I have a hard time getting my head around WHY the gradient in my case must have the size 10x1/ 10x2 and how I need to set them to work properly. I mean I can't get a deriative for a variable thats not in the formula?! Do I just pass in an empty vector for that specific deriative?

I'm really not sure If im just thinking way to complicated (again). I'm also wondering why I an empty gradient for the inequality constraint won't work?!

as always, any help will be greatly appreciated!