High-order chain rule for vector valued functions

Question

Let $f,g:\mathbb R^n\to \mathbb R^n$ be $C^\infty$ functions. I know that the chain rule applies as $$D(f\circ g)(p)=Df(g(p))\circ Dg(p)\tag{*},$$ but I want to know if we could apply the chain rule again for this composition, that is, let $r:\mathbb R^n\to \operatorname{End}(n,\mathbb R)\subset \mathbb R^{n\times n}$ be given by $p\mapsto D(f\circ g)(p)$. Is there a way of applying the chain rule again to $(*)$ to give a nice looking expression to $Dr(v)$ for some $v\in \mathbb R^n$?

Answer 1

If I understand well, you're looking to find the second derivative of a composition of maps. This is given by

$$D^2(f \circ g)(p)(x_1,x_2) = D^2f(g(p))(Dg(p)\cdot x_1,Dg(p)\cdot x_2) + Df(g(p)) \cdot D^2g(p)(x_1,x_2).$$

Interesting is to understand in details what are the domains and codomains of the various maps involved in this formula.