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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