Limiting a Secant Product

Calculus Level 4

Let

\[\large{f(x) = \lim_{n \to \infty} 2^{-n} \prod_{r=1}^{n} \left( \sec\left(\dfrac{x}{2^{r-1}} \right) +1 \right)}\]

If \(f' \left(\dfrac{\pi}{4}\right)\) can be written as:

\[\large{\dfrac{A}{\pi} - \dfrac{B}{\pi ^2}}\]

for positive integers \(A\) and \(B\), submit the value of \(A+B\) as your answer.

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