The capital delta acts like a "normal" derivative if you want to say so. In eq. 4, you can see the product rule which is one of the most defining features for any sort of calculus. However, I must admit that this summary will take me a lot of time to digest...
Yes, it obeys enough algebraic laws that calling it a derivative is useful.
But I don't think there's any underlying notion of small changes to something continuous. It is not the slope of some smooth function.
It involves defining a set of operators — like functors in CS, operators take one function and return another — which obey a modified form of the product rule for derivatives D[f*g] = g*Df + f*Dg. These operators are used to make the analytic continuation of divergent series consistent; because they are defined in terms of functions that cannot be calculated directly from their definition (hence analytically continued), they are "alien".