A general proof of the Chain Rule Let f and g be
Chapter 3, Problem 103(choose chapter or problem)
A general proof of the Chain Rule Let f and g be differentiable functions with h1x2 = f 1g1x22. For a given constant a, let u = g1a2 and v = g1x2, and define H1v2 = c f 1v2 - f 1u2 v - u - f _1u2 if v _ u 0 if v = u. a. Show that lim vSu H1v2 = 0. b. For any value of u show that f 1v2 - f 1u2 = 1H1v2 + f _1u221v - u2. c. Show that h_1a2 = lim xSa a 1H1g1x22 + f _1g1a222# g1x2 - g1a2 x - a b. d. Show that h_1a2 = f _1g1a22g_1a2.
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