I would like to adapt Abramowitz and Stegun 5.1.27 formula to my problem.
Equation 5.1.27 states:
$$\frac{d^n}{dz^n}\left[e^z\,E_1(z)\right]=\frac{d^{n-1}}{dz^{n-1}}\left[e^z\,E_1(z)\right]+\frac{(-1)^n\,(n-1)!}{z^n}$$
In my case I have:
$$e^{az}\,E_1(az)$$
so to have the $n$-th derivative I am thinking of doing:
$$\frac{d^n}{dz^n}\left[e^{az}\,E_1(az)\right]=a^n\frac{d^n}{dx^n}\left[e^{x}\,E_1(x)\right]$$
Is it right? Can I just scale by the prefactor? Has someone any idea to prove or disprove it? And maybe help me with the right derivation in my case?