It's easy to see that the derivative of $detA$ is $adj(A)^T$ applied to $A(t)$ we have $adj(A(t))^T$ and the derivative of $A(t)$ is simply $A'(t)$.
Now "multiplying" all this together with inner product ($tr$) gives $tr(adj(A)A'(t))$ which is what was desired.
I haven't seen this proof so far so I'm wondering if it's correct and possibly why it's not used. Is it because inner product isn't unique? Is the chain rule correct in those cases?