f(x) is a function of x, but f'(x) is the derivative with respect to what seems to make sense. In terms of a-level f'(x) pretty much means dy/dx, but you should realise that d(f(x))/dx = f'(x) doesn't require a y.
As for the post, it depends what you are differentiating with respect to but I assume it is p (i.e. do you want dq / dp ?). Btw, q = (100 - p) / 5.5, then also q = 100/5.5 - p/5.5, now if I said q = y, and p = x, then y = 100/5.5 - x/5.5. Can you find dy/dx?