He was so lucky!

Calculus Level 4

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

(f×g)=f×g(f \times g)' = f' \times g'

However, he was lucky to get the correct answer. The function ff that he used was f(x)=ex2f(x) = e^{x^{2}}. The domain of g(x)g(x) is the interval (12,)(\frac{1}{2}, \infty) with g(1)=eg(1) = e.

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

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