A local analysis textbook I have used has the following exercise:
Let $X$ be a finite-dimensional, $Y$ a separable Banach-space, $f\colon X\rightarrowtail Y$ any function. Show that $f'$ is Borel.
I have no idea how to approach this. I looked at the proof of Rademacher, but even that only gives Lebesgue-measurability as far as I can see, and this has no assumptions on the function at all. Any help would be appreciated.