Pointwise complexity of the derivative of a computable function

We explore the relationship between analytic behavior of a computable real valued function and the computability-theoretic complexity of the individual values of its derivative (the function’s slopes) almost-everywhere. Given a computable function f, the values of its derivative $$f'(x)$$ , where they are defined, are uniformly computable from $$x'$$ , the Turing jump of the input. It is known that when f is $${\mathcal {C}}^2$$ , the values of $$f'(x)$$ are actually computable from x. We construct a $${\mathcal {C}}^1$$ function f so that, almost everywhere, $$f'(x)\ge _T x'$$ . Although the values $$f'(x)$$ at each point x cannot uniformly compute the corresponding jumps $$x'$$ of the inputs x almost everywhere for any $${\mathcal {C}}^1$$ function f, we produce an example of a $${\mathcal {C}}^1$$ function f such that $$f(x)\ge _T \emptyset '$$ uniformly on subsets of arbitrarily large measure, effectively (using the notion of a Schnorr test). We also explore analogous questions for weaker smoothness conditions, such as for f differentiable everywhere, and f differentiable almost everywhere.