Continuing from my previous question, let $f : S \subseteq \mathbb{R} \to \mathbb{R}$ be differentiable on $S$, let $x \in S$, and consider the derivative of a composite of its derivative, evaluated at a specific point:
$$ \frac{d}{dy}\bigg|_{y = x}\big[[f'(y) - f'(x)]^2\big] $$
The counterexample in the answer to my previous question makes use of a function which is not the derivative of another function. With this further restriction, is the above expression defined? It seems like the discussion here of the maximum degree of discontinuity of a derivative may be relevant, but I can't seem to find the answer to my specific question there.
No, it's not, for the same argument as in your previous question; in fact, this question is in no way different from the previous one if you take $F = f' : S \to \Bbb R$ and apply the previous reasoning to $F$ instead of $f$ (since it is not necessary that $F$ be differentiable).