Let us give the power series
f(x) = c0 + c1 x + c2 xx + c3 xxx +....
From this define a modified series with capital F
F(x) = c0 + c1 x +c2 x(x+h) + c3 x(x+h)(x+2h) +...
This is really a function of two variables, but concéntrate on the x variable.
The derivative of the original series is Df
Df = c1 + 2 c2 x +...
We call now the transformation of the derivative series TDF
The point of interest is now
F(x+h) - F(x) = h TDF(x+h)
We get the right hand side by first calculating the transformation of the derivative function Df, to get TDF and then this series
is evaluated not at x but at x+h.
These operations are not commutative.
This shows the connection between the continium and the discrete in
concrete form. Hopefully the result is correct.
If h tends to zero one just gets Df=Df