Thank you very much for this very insightful answer. 1. Can you tell me why $(M^{-1})^\mu_\nu \frac{dM^\nu_\mu}{dx^\lambda} = \frac{d \text{tr} \ln M}{dx^\lambda}$, and why $\frac{d\ln \det M}{dx^\lambda} = \frac{d \text{tr} \ln M}{dx^\lambda}$? 2. Furthermore, I don't understand why you have edited all the partial derivatives in my original formulation of the question into direct derivatives. I think the partial derivative version is correct?