[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.