👉 Differences, often denoted as Δx or x₁ - x₂, measure the change in a quantity from one state (x₁) to another (x₂). It quantifies how much x₂ differs from x₁, capturing both the magnitude and direction of the change. Unlike differences (x₂ - x₁), which reverse the order, differences are symmetric (Δx = -(x₂ - x₁)), meaning Δx = -Δx reflects the same change in both directions. This concept is foundational in calculus, where derivatives (limits of differences over infinitesimal changes) quantify instantaneous rates of change, enabling analysis of functions' behavior like maxima, minima, and slopes. It also underpins error analysis in approximations, where small differences approximate changes in quantities, and is critical for modeling real-world phenomena like motion or growth rates.
