Here is the confusing calculation, if you care (I will do this with general $n$): We use all the identifications and notations from earlier. The map $f : \Bbb R^n \to \Bbb R$ gives derivative $Tf : T\Bbb R^n \to T\Bbb R$, $Tf(x, u) = (x, Df(x)u)$. Taking derivative again, and using $T^{(2)} \Bbb R^n = \Bbb R^n \times T_0 \Bbb R^n \times T_0 \Bbb R^n \times T_0T_0 \Bbb R^n$, I get $Tf(x, u_1, u_2, v) = (x, Df_x(u_1), Df_x(u_2), D^2f_x(v, u_2))$. Now, using
$$T^{(3)} \Bbb R^n = (\Bbb R^n \times T_0\Bbb R^n) \times (T_0 \Bbb R^n \times T_0T_0\Bbb R^n) \times (T_0 \Bbb R^n \times T_0 T_0 \Bbb …