For discontinuous functions, operator {difference operator} can find finite difference between (n+1)th term and nth term: y(n + 1) - y(n). First-order difference operator symbol is uppercase Greek letter delta. Second-order difference operator symbol is uppercase Greek letter delta squared.
Vanishingly small increment or infinitesimally small interval can have symbol dx {differential}. dx = (x + delta_x) - x, as delta_x approaches zero. For function, df(x) = f(x + delta_x) - f(x), as delta_x approaches zero. Therefore, (f(x + dx) - f(x)) / ((x + dx) - x) ~ df(x) / dx or f'(x) and f(x + dx) ~ f(x) + f'(x) * dx ~ f(x) + df(x).
Variable partial derivative times variable differential is variable change {exact differential}: (Df(x,y) / Dx) * dx = (x change), where f(x,y) is a two-variable function, D is partial derivative, and dx is differential. For all variables, sum exact differentials. For two variables, (Df(x,y) / Dx) * dx + (Df(x,y) / Dy) * dy = (x change) + (y change). First-order differential equation can use differentials.
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Date Modified: 2022.0225