He was so lucky!

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 $\alpha + \beta$.