The very recently introduced idea of Quasi Differential Quotient is presented and discussed, starting from the definition and easy examples and comparing the idea with other generalized differentiation theories. The talk has the ambitious aim of explaining why this new mathematical tool is worth exploring, as it has reasonable properties that make it practical and useful in applications. An application in the field of optimal control theory is discussed, obtaining a second order pontryagin maximum principle for a control affine problem with non-smooth fields. This is done with the help of Quasi-Differential Quotients and the idea of a set-valued Lie Bracket for Lipschitz fields.