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.