On the real differential of a slice regular function

In this paper we show that the real differential of any injective slice regular function is everywhere invertible. The result is a generalization of a theoremproved by G. Gentili, S. Salamon and C. Stoppato and it is obtained thanks, in particular, to some new information regarding the first coefficients of a certain polynomial expansion for slice regular functions (called spherical expansion), and to a new general resultwhich says that the slice derivative of any injective slice regular function is different from zero. A useful tool proven in this paper is a new formula that relates slice and spherical derivatives of a slice regular function. Given a slice regular function, part of its singular set is described as the union of surfaces onwhich it results to be constant.