Suppose that $G$ is a connected Lie group of unitary matrices and $U(t)\in G$ depends continuously differentiable on a real parameter $t$ and has no real eigenvalue -1. Then the principal value of the logarithm $W(t):=\log U(t)$ is also continuously differentiable in $t$. 


If $G$ is abelian, we may conclude (with dot denoting differentiation by $t$) the formula $$\dot W(t)=U(t)^{-1}\dot U(t).$$
I am looking for an explicit formula (with proof) generalizing this that expresses $\dot W(t)$ in terms of $U(t)$ and $\dot U(t)$ when $G$ is nonabelian.



A matrix derivative
The comment by Vít Tuček leads to the formula
$$\frac{d}{dt}\log(1+A(t))=\int\_0^1 (1+sA(t))^{-1}\dot A(t)(1+sA(t))^{-1} ds.$$
This can be verified by Taylor expanding both sides if the norm of $A(t)$ is less than 1, and follows in general by analytic continuation when no real eigenvalue of $A(t)$ is $\le -1$. (The proof in <https://math.stackexchange.com/a/2085547/27752> only covers the first situation.)


