In Riemannian Geometry by Do Carmo, he defines the differential of a function between manifolds as the following:
My confusion is the following; he defined the differential as a map between the tangent spaces however it is unclear how $\beta’(0)$ could possibly act on a smooth function as $\alpha’(0)$ defined previously did. Before, alpha was a sum of differential operators however here beta would be a column matrix and from what I can tell wouldnt even have any differential operators as entries. I understand that this proposition simply serves to introduce the differential but it doesn’t seem consistent with previous definitions