Proving: inequality 1 / lnx-1 / (x-1)

Let f (x) = 1 / lnx-1 / (x-1) - 1 / 2, X ∈ (1,2). Then f '(x) = - 1 / [x (LNX) ^ 2] + 1 / (x-1) ^ 2 = - G (x) / [x (x-1) ^ 2 * (LNX) ^ 2], where g (x) = (x-1) ^ 2-x (LNX) ^ 2, G' (x) = 2 (x-1) - (LNX) ^ 2-2lnx, G "(x) = 2-2lnx / X-2 / x = 2 (x-lnx-1) / x, let H (x) = x-lnx-1, X ∈ (1,2

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