I am trying to compute or approximate $$\frac{d}{dt}||e^{tA}||_F^2$$ where $||\cdot||_F$ denotes the Frobenius norm. $A$ is square, but it is not necessarily symmetric or diagonalizable. However, $A$ is always negative semidefinite.
I will accept an answer that shows how to either write down $\frac{d}{dt}||e^{tA}||_F^2$ in a useful form or gives a good approximation.