On differentiability of Sobolev functions with respect to the Sobolev norm

We study connections between the $W_p^1$-differentiability and the $L_p$-differentiability of Sobolev functions. We prove that, $W_p^1$-differentiability implies the $L_p$-differentiability, but the opposite implication is not valid. The notion of approximate differentiability is discussed as well. In addition, we consider the $W_p^1$-differentiability of Sobolev functions ${\text{cp}}_p$-almost everywhere.