Do the chain rules for matrix functions hold without commutativity? Linear and Multilinear Algebra

This paper shows that the commutativity condition [A(t), A′(t)] = 0 is often not necessary to guarantee the chain rules for matrix functions: ddtf(A(t)) = A ′(t)f ′(A(t)) and ddtf(A(t)) = f ′(A(t))A′(t), where A(t) is a square matrix of differentiable functions and f is an analytic function. A further question on the chain rules is presented and discussed