I am trying to prove a claim made on Brezis' "Functional Analysis, Sobolev Spaces and PDE's", that if $I=(a,b)$ is an interval (bounded or unbouded), $u\in C^1(I)\cap L^p(I)$ and $u' \in L^p(I)$ (classical derivative), then $u \in W^{1,p}(I)$ and the weak derivative coincides with the classical one.
My attempt: if $\varphi \in C_c^1(I)$, then in particular $u\varphi \in C^1(I)$. Then, differentiating the product we get $$u\varphi'=(u\varphi)'-u'\varphi$$ Integrating, we get $$\int_I u\varphi'=\int_I(u\varphi)'-\int_Iu'\varphi $$ But I cannot use the Fundamental Theorem of Calculus on the second integral to write $\int_a^b(u\varphi)'=u(b)\varphi(b)-u(a)\varphi(a)$, because these functions are defined only on $(a,b)$. I have a feel that somehow the second integral must vanish, but I cannot prove it.