Question

求大佬看看微积分3题的(2)

Answer 1

f(x+y) = f(x)f(y)
x=y=0
=>f(0) = 1
f'(0)
=lim(h->0) [f(h)- f(0) ]/h
=lim(h->0) [f(h)- 1 ]/h
=lim(h->0) [1+hg(h)- 1 ]/h
=lim(h->0) g(h)
=lim(h->0) g(h)
=1
----------------------
f'(x)
=lim(h->0) [f(x+h) - f(x) ]/h
=lim(h->0) [f(x).f(h) - f(x) ]/h
=lim(h->0) f(x)[f(h) - 1 ]/h (0/0)
=lim(h->0) [1+xg(x) ].[f(h) - 1 ]/h
=[1+xg(x) ] .lim(h->0) [f(h) - 1 ]/h
=[1+xg(x) ] .f'(0)
= 1+xg(x)