Existence of directional derivatives of the gradient of $W^{1,\infty}\cap W^{2,2}$ functions

Question

According to Rademacher's Theorem, any Lipschitz function $f:\mathbb{R}^n \supset E \to \mathbb{R}$ is differentiable. In the first step of the proof, Rademacher shows that the directional derivative $\partial_v f(x)$ exists a.e. in $E$ for any $|v|=1$. I was wondering whether the regularity $W^{1,\infty}(E) \cap W^{2,2}(E)$ sufficies to deduce that any directional derivatives of $\nabla u$ exist a.e. in $E$, i.e. $\partial_v \nabla u$ exists a.e. for every $|v|=1$. It is clear that boundedness of $\nabla ^2 f$ would guarantee the existence of $\partial_v \nabla u$. Intuitively this should (unfortunately) not hold true in the case where only $f \in W^{1,\infty}(E) \cap W^{2,2}(E)$, but maybe I am wrong. Maybe one could try to approximate $f$ with smooth functions, but I am not sure whether this could work, and if so, how. I appreciate any remarks!