The fundamental dispute was not over the notation, but over whether the notation was misleading. I.e., is df/dx = (df/dz)(dz/dx) a real description of what is h

The fundamental dispute was not over the notation, but over whether the notation was misleading. I.e., is df/dx = (df/dz)(dz/dx) a real description of what is happening, and a valid calculus, or just a coincidence that sometimes won't work? Newton had a variety of arguments in favor of why his calculus worked, only some based on infinitesimals. Liebniz just asserted without proof that infinitesimals were fine, used them, and got good results. Additionally, there were a variety of religious arguments mixed into this, most of which are lost in the sands of uninteresting history. I.e., disputes over whether god (the perfect watchmaker) periodically wound his watch or whether he build a perfect watch that never needed winding. For instance, Liebniz believed in noninteracting dualism, but got set up the world so perfectly that the mind and body agreed. (I'm a mathematician and only minimally familiar with these - they have been discarded by history.) These religious arguments did play a role in the debate over infinitesimals, and acceptance of divine coincidences like Liebniz notation - unjustified though it may be - working perfectly all the time.