Difference and Difference Quotient. Part II

The articles [8], [1], [4], [2], [3], [5], [7], [12], [13], [6], [9], and [10] provide the notation and terminology for this paper. We follow the rules: h, r, r1, r2, x0, x1, x2, x3, x4, x5, x, a, b, c, k denote real numbers and f , f1, f2 denote functions from R into R. Next we state a number of propositions: (1)1 ∆[f ](x, x+ h) = ( ~ ∆h[f ])(1)(x) h . (2) If h 6= 0, then ∆[f ](x, x+ h, x+ 2 · h) = (~ ∆h[f ])(2)(x) 2·h2 . (3) ∆[f ](x− h, x) = (~ ∇h[f ])(1)(x) h . (4) If h 6= 0, then ∆[f ](x− 2 · h, x− h, x) = (~ ∇h[f ])(2)(x) 2·h2 . (5) ∆[r f ](x0, x1, x2) = r ·∆[f ](x0, x1, x2). (6) ∆[f1 + f2](x0, x1, x2) = ∆[f1](x0, x1, x2) + ∆[f2](x0, x1, x2).