# Coincidence of standard derivative and weak derivative

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 $p$?

• I think you are right, thanks. – Asaf Shachar Jan 19 '17 at 12:25
• For completeness, I have posted it as an answer. – gerw Jan 19 '17 at 18:38