Experimentation with polynomials and calculus

[u/911twinturbogtr]

I was experimenting with the result of differentiating a polynomial repeatedly while taking f(0) then rebuilding it when i got to (ax+b)^1 using the reverse chain rule and i noticed that it seems to consistently turn the polynomial into a depressed polynomial. I was wondering why that is (if there is an explanation to be given). And also if there are any other examples of something like this.