Let $A, B : \text{Sym}(d) \rightarrow \text{Sym}(d) $, where $\text{Sym}(d)$ is the set of $d \times d$ symmetric matrices. I am interested in computing the second order differential of $F(X) = A(X) B(X)$, more precisely $D^2 F(X)[U, U]$ for some $U \in \text{Sym}(d)$. I am currently using the following formulas
$$ DF(X)[U] = DA(X)[U] \; B(X) + A(X) \; DB(X)[U] $$
$$ D^2 F(X)[U, U] = D^2A(X)[U, U] \; B(X) + 2 DA(X)[U] \; DB(X)[U] + A(X) \; D^2 B(X)[U,U] $$
However, I am not completely confident in them (especially since I am not familiar with tensors). I would appreciate if you could help me with a reference for these.