in my research I ended up with a term of the following form: $$ C(x)^\prime:C^{-1}(x)^\prime $$ In my case the matrix function $C(x)\in\mathbb{R}^{3\times 3}$ and is always s.p.d. so we can rewrite the above as $$ -\text{trace}(C^{-\frac{1}{2}}C^\prime C^{-\frac{1}{2}}C^{-\frac{1}{2}}C^\prime C^{-\frac{1}{2}}) = \|C^{-\frac{1}{2}}C^\prime C^{-\frac{1}{2}} \|_F^2. $$
Actually I want to know if $C^{-\frac{1}{2}}C^\prime C^{-\frac{1}{2}}$ can be written as derivative of a matrix function, i.e. $F(C)^\prime$.
I know that if $C$ and $C^\prime$ commutes then $F=\log(C)$.
However, in the non-commuting case the derivative of the matrix logarithmn is quite ugly. However I saw something on logarithmic derivatives which share some properties, see here.
Maybe somebody has an idea or a reference for this logarithmic derivatives.