first principle

mqlvkson

我想問點用first principle 黎搵e^x 既 derivative

Naozumi

呢個問題歸結為 (e^h - 1)/h = ? as h->0
Put y = e^h - 1,
then h = ln (y+1)

Now
(e^h - 1)/h = y/ln(y+1)
                    = 1/ [ln (y+1)]/y
                    = 1/ ln[(y+1)^(1/y)]

as h->0, y->0
also, lim [(y+1)^(1/y)] = e
so lim (e^h - 1)/h = 1

有左呢條式用first principle計 derivative of e^x就 no problem

呢度我assume左  f(x) = ln x 係continuous function,唔再證
同埋用左  e的另一個變種definition, e = lim(x-> infinity) (1 + 1/x)^x
而家put y = 1/x 變左e = lim(y-> 0) (1 + y)^(1/y)

[ 本帖最後由 Naozumi 於 2009-3-29 01:15 AM 編輯 ]