\[ \frac{d}{dx} \left(f \cdot g\right) = f' \cdot g'\]

Let's denote the above equation the *fake product rule*. Although it is generally the wrong way to differentiate the product of two functions, there are non-constant functions \(f\) and \(g\) for which it holds true.

Let \(f(x) = x^2\) and \(g(x)\) be the functions such that the fake product rule holds true.

If \(g(1) = 1\) and \(g(5) = \frac19\), find \(g(3) - g'(3) - \big[f'(4) \cdot g'(4)\big]\).

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