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Let $f:\mathbb{R}^n \to \mathbb{R}$ be in $W^{1,p}(\mathbb{R}^n)$ and differentiable (in the classical sense) almost everywhere.

Is it true that the standard derivative and the weak derivative conicide?

When $p>n$ this is a corollary from theorem 4.9 ("LECTURES ON LIPSCHITZ ANALYSIS"- by Heinonen; In fact, in that case $f \in W^{1,p}$ implies that $f$ is differentiable almost everywhere).

What happens for other values of $p$?

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    I think you are right, thanks. – 2017-01-19
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    For completeness, I have posted it as an answer. – 2017-01-19

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This follows from the ACL characterization of Sobolev spaces, see https://en.wikipedia.org/wiki/Sobolev_space#Absolutely_continuous_on_lines_.28ACL.29_characterization_of_Sobolev_functions.