Non- (quantum) differentiable $C^1$-functions in the spaces with trivial Boyd indices

If E is a separable symmetric sequence space with trivial Boyd indices and ${\text{cC}}^E$ is the corresponding ideal of compact operators, then there exists a $C^1$-function $f_E$, a self-adjoint element $W\in {\text{cC}}^E$ and a densely defined closed symmetric derivation $\delta$ on ${\text{cC}}^E$ such that $W\in Dom\delta$, but $f_E(W)\mathrm{\notin }Dom\delta$.