Certainly there was no clear math/physics separation at the time (and the term physics as currently understood didn't even exist), but the drive to develop the math necessary to solve real-life problems was certainly there, and separate from the drive to do pure mathematics

Except that the problems in question (explaining the motion of the planets and so on) would not have been considered "real-life problems" back then; rather, they would have been considered "abstract philosophical speculations" that would have carried the same kind of stigma among "practical men" that "pure mathematics" does today.

And it took a long time before the d/dx notation was properly formalized.

I think you mean "justified"; if there was one thing Leibniz was good at, it was formalizing!

As for the Noether's theorem, it was inspired by Einstein proving the energy-momentum tensor conservation in General Relativity, without realizing that it was a special case of a very general principle.

According to your Wikipedia link, it represented the solution to a physical problem in its own right: a paradox wherein conservation appeared to be violated in GR.