A student forgot the product rule for differentiation and made the mistake of thinking that

\[(f \times g)' = f' \times g'\]

However, he was lucky to get the correct answer. The function \(f\) that he used was \(f(x) = e^{x^{2}}\). The domain of \(g(x)\) is the interval \((\frac{1}{2}, \infty)\) with \(g(1) = e\).

If the value of \(g(5) = \alpha e^{\beta}\), where \(\alpha\) and \(\beta\) are integers, find \(\alpha + \beta\).

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