I'm a bit new to optimization methods, and I noticed in some definitions like Gauss-Newton wikipedia page the cost derivatives (Jacobian or other) are w.r.t the parameters themselves. However, in other contexts such as the answer to another question here, the optimization derivatives are defined w.r.t. to the infinitesmal change evaluated at 0. Conceptually, they seem similar, however, from an implementation standpoint there is a large difference. For example given the context of the problem of the second link, evaluating derivative of a function of rotations w.r.t. axis-angle is a cumbersome equation, while evaluating the derivative w.r.t. to the infinitesmal axis-angle change is as simple as the generating function in the Lie space.
More fundamentally, is there a difference between optimizing a cost w.r.t. to some variables and optimizing a cost w.r.t. to the perturbation of some variables or is it just a math trick exploited depending on optimization context?