I don't think xTx is is a great example for the demonstration of matrix calculus techniques, because it is a scalar. So, by definition, its derivative with respect to a vector v is going to be the gradient -- a vector of that scalar differentiated by each component of v. This is such a simple case it's not really necessary to introduce matrix calculus at all.

It's when you take derivatives of vectors & matrices by other vectors & matrices that things get "interesting".