Consider the function $V:\mathbb{R}\to\mathbb{R}$ given by
$ V(t)=\|I - e^{At}\|^2 $
where $I$ is the identity matrix and $A$ is a square matrix. The norm is the Euclidean norm on $M_n(\mathbb{R})$:
$\|X\|=\sqrt{\lambda_{max}(X'X)};\ X\in M_n(\mathbb{R})$
that is induced by the matrix norm $\|x\|^2=x'x$ on $\mathbb{R}^n$.
I want to calculate the derivative $\frac{dV(t)}{dt}$. Is this possible in any way?
Note: Maybe the use of the Frobenius norm would facilitate things a bit but I wouldn't prefer it as it is not an induced norm (by some matrix norm).