I would like help proving this elementary result:
Let $f\in L^{1}_{loc}(a,b)$. Let $x_0 \in (a,b)$ Let $F(x)=\int^{x}_{x_0} f$. Then $F'=f$ in the sense of distributions.
i.e How do I show $\langle F',\phi\rangle=\langle f,\phi\rangle$?
All I know is that $\langle F',\phi\rangle = -\langle F, \phi'\rangle$. I do not see how the right hand side is equal to $\langle f,\phi\rangle$