Consistency of definition of weak derivative with classical derivative I know the definitions of both weak and classical derivative. But I am trying to see the classical derivative as a weak derivative. When we have
$\int f' \varphi = -\int f\varphi'$ for all $\varphi\in C^\infty$ with compact support. Is this definition consistent with the one for classical derivative, I mean, if $f\in C^1$ how can we obtain $f'$(classical) from the def. above?

Integration by parts:
$$
\int_{-\infty}^\infty f\phi'\,dx = \lim_{\substack{a\to-\infty\\b\to\infty}} \left[ f\phi \right]_a^b - \int_{-\infty}^\infty f'\phi\,dx = - \int_{-\infty}^\infty f'\phi\,dx
$$
since $\phi(a) = \phi(b) = 0$ if $|a|$ and $|b|$ are sufficiently large. (Remember that $\phi$ has compact support.)
