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1. Suppose g(x, f) = f^{3}(x) - x = 0

Then 0 = dg = 3f^{2}df - dx that is

We get

2. g(x, f) = xf(x) - 1 = 0

and

Since f(x) = 1 / x, we get (1 / x)' = - (1 / x^{2})