2 15 points A function z = f(x, y) is defined implicitly viaa + y' - 22 + 12x - 82 = 4. Apply implicit differentiation to find am a ay 19 and "2 and evaluate them at the given points (1, 1, 1) and (1, 1, -5). You can directly apply the formulas for a suitably chosen F: az Fx az Fy ax ay F . Or, you can derive the above formulas by yourself, as done in the following. Introduce the function F(u, v, w) = u' tv' -2w + 12u -8w, u = x, v= y, w = z and treat u, v, was independent, then apply chain rule (this is a multi- variable chain rule, although here you only need to differentiate wit to r, y): The given implicit function means F(u, v, w) = u + v -2w' + 12u - 8w =4. Differentiate witz, y on both sides (you need chain rule on the left-hand side), you get du du + Fo da Fx = Fu dx aw - 0, + Fu Or Fy = Fu dy du + Fo dy du + Fu dy 1 = 0 Note w = 2(@, y), thus you use partial derivative notations for w. Certainly you can use partial derivatives for u, v as well, but since they are single variable functions of , standard derivative notation should be just fine (and simpler). Apply the relationship that I, y are independent, thus de = = 0 and $4 dy " = 0, you then get the above listed formulas (certainly you need to replace u, v, w by , y, Z. At a more advanced level, you can skip the stepping stone u, v, w and directly use , y, 2, but for beginners, it is easier to understand the chain involved in the whole process with the stepping stone). Now input what you eventually get: (you answers should be in term of the original variables r, y, z) Oz a b = type your answer. 8+ 6' where the functions represented by a, b should be a = type your answer. "If the answer contains a sum, input the constant term before the non-constant term. Use same input rules as before.) Oz where bis the same as above, and the function represented by c should be c = type your answer.. ay 8+ 6 ' Evaluations: (when not an integer, input most simplified fraction) ax oz (1, 1, 1) = type your answer. ay "z (1, 1, 1) = type your answer. Oz - (1, 1, -5) = type your answer. ax Oz (1, 1, -5) = type your