  It is known that there exist mathematical problems of practical relevance
which cannot be computed on a Turing machine. An important example is the
calculation of the first derivative of continuously differentiable functions.
This paper precisely classifies the non-computability of the first derivative,
and of the maximum-norm of the first derivative in the Zheng-Weihrauch
hierarchy. Based on this classification, the paper investigates whether it is
possible that a Turing machine detects this non-computability of the first
derivative by observing the data of the problem, and whether it is possible to
detect upper bounds for the peak value of the first derivative of continuously
differentiable functions. So from a practical point of view, the question is
whether it is possible to implement an exit-flag functionality for observing
non-computability of the first derivative. This paper even studies two
different types of exit-flag functionality. A strong one, where the Turing
machine always has to stop, and a weak one, where the Turing machine stops if
and only if the input lies within the corresponding set of interest. It will be
shown that non-computability of the first derivative is not detectable by a
Turing machine for two concrete examples, namely for the problem of computing
the input--output behavior of simple analog circuits and for solutions of the
three-dimensional wave equation. In addition, it is shown that it is even
impossible to detect an upper bound for the maximum norm of the first
derivative. In particular, it is shown that all three problems are not even
semidecidable. Finally, we briefly discuss implications of these results for
analog and quantum computing.
