## differentiation of a monotone function

I got stuck with this problem
Lef f be a monotone function on [a,b] and define $F(x) = \int_a ^x f'(t) dt$.
I need to show that $F'(x)=f'(x)$ for almost every x in [a,b].
if f' was continuous then this will be simply the fundamental theorem but all I know about f is that it is non negative (and integrable a.e. because f is monotone).