So I've seen this joke going around about people using derivatives as fractions and I can kind of see the reasoning behind it since dy/dx is still just two expressions being divided with dy= (x +dx)- dx
and with
dx = (x + very small number) - x
and those expressions can still be manipulated individually but they probably wouldn't yield an insight like derivatives do. However, my question is where in the world of mathematics is this being done? I would appreciate a link to an example where this is being done or at least a general point to what subject this is being done in.