The notation may be making the problem more complicated than it appears??:
I don't think you need to know anything more than $\displaystyle \frac{d}{dx}ax^n = anx^{n-1}$
Here is an image from your attachment:
(The $\displaystyle \displaystyle \alpha_i$ are real coefficients.)
I think there is a typo in the expression on the right-hand side of the equation.
$\displaystyle \displaystyle {{d}\over{dz}}\sum_{i=0}^n{{\alpha_i z^i}\over{i!}}$
$\displaystyle \displaystyle = {{d}\over{dz}}\alpha_0+\sum_{i=1}^n{{d}\over{dz}}{ {\alpha_i z^i}\over{i!}}$
$\displaystyle \displaystyle =0+\sum_{i=1}^n{{(i)\alpha_i z^{(i-1)}}\over{(i)(i-1)!}}$
$\displaystyle \displaystyle =\sum_{i=0}^{n-1}{{\alpha_{i+1} z^{i}}\over{i!}}$