By definition of derivative, #F# is differentiable in #x_0# if the limit:
#lim_(x->x_0) (F(x)-F(x_0))/(x-x_0)#
exists.
Consider now any sequence #{x_n}# such that:
#lim_(n->oo) x_n = x_0#
note that:
#F(x_n)-F(x_0) = int_a^(x_n) f(t)dt - int_a^(x_0) f(t)dt#
and using the additivity of the integral:
#F(x_n)-F(x_0) = int_(x_0)^(x_n) f(t)dt #
Based on the mean value theorem, there must then be a point #xi_n in (x_0,x_n)# such that:
#int_(x_0)^x f(t)dt =(x_n-x_0)f(xi_n)#
so that:
#(F(x_n)-F(x_0) )/(x_n-x_0) = f(xi_n)#
Now, as #xi_n in (x_0,x_n)# and #lim_n x_n = x_0# it follows that necessarily also:
#lim_n xi_n = x_0#
and as #f(x)# converges (really we need here #ƒ(x)# to be continuous) in #x_0# then:
#lim_n f(xi_n) = f(x_0)#
and then:
#lim_n (F(x_n)-F(x_0) )/(x_n-x_0) = f(x_0)#
and because #{x_n}# is arbitrary:
#lim_(x->x_0) (F(x)-F(x_0))/(x-x_0) = f(x_0)#
which is basically the proof of the fundamental theorem of calculus.
