Derivative of an exponential mapping with respect to a scalar This is a follow up question to one I've asked previously.
How can I take the derivative of this exponential mapping / what is the solution?

Let
$$A (x) := \begin{bmatrix}0&0\\4x&1\end{bmatrix}$$
and let $x$ be a scalar. Then what is the solution to $\frac{\partial}{\partial x} ( e^{A(x)})$?

If there is an analytical solution, that is preferable, but numerical approximation are will work too. Thank you.

$A^n=A$ for all $n\geq 1$ is easy to check.
Therefore, $$e^A=\sum_{n=0}^\infty \frac{1}{n!} A^n=I+A \sum_{n=1}^\infty \frac{1}{n!}.$$ Also, $$\frac{\partial}{\partial x}A = \begin{bmatrix}0&0\\4 &0\end{bmatrix},$$ which gives $$\frac{\partial}{\partial x} e^A=\frac{\partial}{\partial x}A \sum_{n=1}^\infty \frac{1}{n!}=(e-1)\begin{bmatrix}0&0\\4 &0\end{bmatrix}$$
