(x˄n)▽x = ｢((x+h)˄−x˄n)/h :h⨠0｣
Binomial Expansion
(x+h)˄n
= ｢n!/((n−k)!·k!)·x˄(n−k)·h˄k Σk=0,n｣
(x+h)˄n − x˄n
= ｢n!/((n−k)!·k!)·x˄(n−k)·h˄k Σk=1,n｣
((x+h)˄n−x˄n) / h
= ｢n!/((n−k)!·k!)·x˄(n−k)·h˄(k−1) Σk=1,n｣
｢((x+h)˄n−x˄n)/h :h⨠0｣
= ｢n!/((n−k)!·k!)·x˄(n−k)·h˄(k−1) Σk=1,1｣
┅
All terms approach 0 except the term where k=1.
┅
= (n!/((n−1)!·1!))·x˄(n−1)·h˄(1−1)
= n·x˄(n−1)
