After playing around with the limit definition of the derivative for higher order derivatives, I noticed the following odd relationship to determine it for an nth order derivative:
Let $F^n=f(x+nh)$ (is there a way to write this properly as an operator on $f(x)$?), then
Expanding the middle equality gives it in terms of $f(x+nh)$s. Notice that the inside of the bracket on the RHS is equal to $f'(x)$ (sort of). I have actually proven that this is indeed true using repeated use of L'Hopital's law and proof by induction, but I am unsatisfied. The result in the form above seems almost magical, and makes me think there is a very elegant reason why. Can someone explain why? Based on the result, I suspect umbral calculus gives a nice explanation, but I do not know a lot about it. So can someone explain why the above has such an elegant form?
NOTE: I realise that this is similar to this question, but that question is asking about if it's true; I already know it's true, but I'm asking why it's true based on the magical form above. Also, I don't care that this evaluates certain derivatives that shouldn't exist (but does correctly evaluate those that do), so do not worry about that.