We continue the analysis in [F. Hansen, and J. Tomiyama, Differential analysis of matrix convex functions. Linear Algebra Appl., 420:102--116, 2007] of matrix convex functions of a fixed order defined in a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus. We amend and improve some points in the previously given presentation, and we give a number of simple but important consequences of matrix convexity of low orders.