Quadratic APN vectorial functions are under intense scrutiny due to their role, e.g., in the big APN problem. Recently, a new tool has emerged to investigate their differential properties: the ortho-derivative. We present new results about this object. We first generalize it as a family of functions that can be defined for any quadratic function, even if not APN. We highlight a relation between the preimage sets of the ortho-derivative and the set of bent components, and between the ortho-derivative and some EA-invariants recently introduced by Kaleyski. We also show it is possible to reconstruct a quadratic function given its ortho-derivative. In the APN case, we prove that its algebraic degree is always at most equal to n−2 using a previously unknown relation between the ortho-derivatives and cofactor matrices.