I got stuck with this problem

Lef f be a monotone function on [a,b] and define $\displaystyle F(x) = \int_a ^x f'(t) dt$.

I need to show that $\displaystyle F'(x)=f'(x)$ 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).