There is a really good paper on better notations for differentiation https://arxiv.org/abs/1801.09553 in such a way that algebraic manipulations of the "dx" etc have meaning. The main proposition is to stop calling the second derivative of y along x d^2y/(dx)^2 since this is completely wrong if you want to manipulate things algebraically. In this notation the exponent is denoting function composition and not multiplication.
Memories of classical mechanics lectures reading a scruffy blackboard from 50 feet away trying to work out whether that’s two Newton-style differentiation dots or only one...
