# Fake product rule

Calculus Level 3

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

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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