By writing it out in index notation (personal preference), the first equation is simple application of the product rule (see Einstein notation)
\begin{align}
\nabla_{A_{ij}} \delta_{kl} A_{km}B_{mn} A^T_{np} C_{pl} = \newline
\nabla_{A_{ij}} A_{km}B_{mn} A_{pn} C_{pk} = \newline
(\nabla_{A_{ij}} A_{km})B_{mn} A_{pn} C_{pk} + (\nabla_{A_{ij}}A_{pn}) A_{km}B_{mn}C_{pk} = \newline
(\delta_{ik}\delta_{jm})B_{mn} A_{pn} C_{pk} + (\delta_{ip}\delta_{jn}) A_{km}B_{mn} C_{pk} = \newline
B_{jn} A_{pn} C_{pi} + A_{km} B_{mj} C_{ik} = \newline
C^T\cdot A\cdot B^T + C\cdot A \cdot B
\end{align}
as for the second equation, I've only seen it derived by rewriting $|A|$ in terms of its eigenvalues and doing some tricks or Jacobi's formula. I don't think the two preceding equations give you much to work with here.