Why is $A$ on the left hand side yet $B$ is on the right hand side when we evaluate $ \frac{d}{dt}(e^{tA}Ce^{tB}) = Ae^{tA}Ce^{tB} + e^{tA}Ce^{tB}B$?

Question

I came across the following expression where $A:Y\to Y$, $B:X\to X$, and $C:X\to Y$ with $X$ and $Y$ being Banach spaces: $$ \frac{d}{dt}(e^{tA}Ce^{tB}) = Ae^{tA}Ce^{tB} + e^{tA}Ce^{tB}B. $$

So we have the product rule followed by the chain rule. Regarding the expression on the right hand side, why is $A$ on the left hand side in the first expression, yet $B$ is on the right hand side in the second expression?

Answer 1

For no special reason. $e^A$ and $A$ commute for any matrix $A$.

In any case, $B$ should be to the right of $C$.