Why, for simple nth degree polnomials' finite difference tables, does the nth (constant) difference set, equal the nth derivative

Question

For example with the equation $f(x)=x^4+2x^3+4x^2+2x+1$ the fourth derivative is $f''''(x)=24$ and when you construct a difference table the fourth difference is 24

Answer 1

Because $(x+1)^n-x^n =nx^{n-1}+ \textrm{lower order terms}$. In other words, only the highest degree term matters and for that term the repeated difference coincides with taking derivatives.