[ewanm89]

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.

[jessriedel]

> 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.)

[yorwba]

If by (Δx)^2 you mean the finite difference (x₂-x₁)² = x₂² - 2x₂x₁ + x₁², then it isn't the same as Δ(x^2) = x₂² - x₁².

If you mean the infinitesimal difference, then Δ(x^2) = 2Δx still isn't (Δx)^2.