I prefer Mathematica's notation over Leibniz's, because it is not as ambiguous (dx can also mean d TIMES x, as a simple example, but also dy^2 can mean d(y^2) and (dy)^2)
I'd go stronger than this and say that Leibniz's notation is actively harmful. It is very useful for quickly doing certain kinds of computations, but at the expense of conceptual understanding for students. Obviously, it's fine to use whatever computational aids you want when you understanding things, but most students are taught nothing but this fragile notation.
It's useful for solving (or partially solving for a set of conditions) linear differential equations, say in the fields of mechanics or electromagnetism. One can work with dx and dt as if they were just factors and move them around quite intuitively.
In the same line, it's great at shining a light on the substitution rule for integration.
Given that your point that it can be obscure at first remains valid, I'd walk the middle line of introducing students to the f'(x) notation first; and after the introduction to integrals introduce this notation to them.
Yea, I agree that, in addition to computational speed, the Leibniz gives some reasonable intuition for some formulas. However, it can also give bad intuition in a multi-variate setting, e.g., dxdyx/dzdw.
A good self-check is to see if you can convert from Leibniz notation to a more rigorous one at any given step in the computation and understand that step rigorously. Personally, I find that functional notation (using D as an operator on the space of functions, etc.) to be as simple to use and much more likely to alert me when I'm about to confuse myself.
I would point out it is d because latin typesetters back then often didn't have a greek typeface to print with and it is the closest to the greek letter delta δ. Once one understands it's δx and δy (or Δx and Δy) and today still today most people don't know how to get delta characters on their latin keyboards, then it is easy just to not use d in algebra and use for differential calculus only. Finally (Δx)^2 and Δ(x^2) are the same thing in differential calculus.
> Finally (Δx)^2 and Δ(x^2) are the same thing in differential calculus.
Isn't Δ(x^2) = 2xΔx ≠ (Δx)^2 ? The object Δ(x^2) has one infinitesimals while (Δx)^2 has two, and the number of infinitesimals is conserved. (You can only get finite quantities by taking the ratio of equal numbers of infinitesimals.)
It really isn't that ambiguous in cursive, though. When writing out "dx" as product in cursive, the "d" and "x" characters are written separately. When writing "dx" as an infinitesimal, the characters are connected. Never had any issue with higher degree derivatives, either.
But I see how it can be confusing with printed characters. I guess Leibniz just took ligatures for granted when he came up with his stuff.
A similar notation is Lagrange notation, which is really useful for studying derivatives as a member of a larger class of operators.
The derivative w.r.t. x of f(x) is D_x f in Lagrange notation. It looks a bit like matrix multiplication for a good reason—a matrix is just a representation of a linear operator on a finite dimensional vector space.
