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.

Since $A=A^2$, we can use the formula for idempotent matrices from the Matrix Cookbook, which is valid for any analytic function of $A$ 
$$\eqalign{
 f(A) &= A\,f(1) + (I-A)\,f(0) \cr
    &= e\,A + (I-A) \cr\cr
 df &= e\,dA - dA \cr
    &= (e-1)\,dA\cr\cr
\frac{\partial f}{\partial x} &= (e-1)\,\frac{\partial A}{\partial x}  \cr 
}$$
