What are the implications of resampling the derivative of a signal with a higher frequency than the original signal?
As Laurent said, no amount of interpolation will restore the information that's not in the original $f_s=1\,\text{Hz}$ signal.

It's been discussed in other posts' comments what a suitable digital differentiator looks like; that's not really easy to answer universally. However, it's certainly a high pass filter. So, I'd say: Try to understand your derivate as a filter operation, if possible. Maybe this gives you a suitable approach on how to implement a "good" (by your application's needs) differentiation.

Also, as Laurent said, when you make assumptions on your original signal, you might find a suitable derivative estimator. The typical approach for DSP (and where Nyquist's theorem mentioned above comes form) is to assume that your signal is a finite sum of complex sinusoids, and hence, every digital signal is equivalently representable by its Discrete Fourier Transform. You might find a derivative according to the differentiation theorem of the Fourier transform, if you want to represent your signal as a sum of orthogonal sinusoids, but you might also represent it according to any other base – and derive a differentiation algorithm that fits your needs best.