I don't know if this is what you are looking for, but here are some thoughts.
Differentiability is defined at a point, however, there are some concepts which relate differentiability to more global properties. The first that come to my mind are the Fourier Transform and Lipschitz Continuity.
The Fourier Transform of the derivative of a function at $\xi$ is the Fourier Transform of that function times $\xi$; that is, \widehat{f'}(\xi)=2\pi i\xi\widehat{f}(\xi). In this way, the differentiability of a function is related to decay of its Fourier Transform. The smoother a function is, the faster its Fourier Transform decays near $\infty$.
The idea of Lipschitz Continuity relates a kind of uniform smoothness to the $L^\infty$ norm of the difference of a function and nearby translates. That is, $\|f-T_hf\|_{L^\infty}\le C|h|$ where $T_hf(x)=f(x+h)$ is a translation operator. In this case, $f$ is differentiable almost everywhere and \|f'\|_{L^\infty}\le C.