Answer
$0$
Work Step by Step
$$f'(0)\\=\lim_{x\to 0}\frac{f(x)-f(0)}{x-0}\\=\lim_{x\to 0}\frac{\cos{x}-\cos{0}}{x}\\=\lim_{x\to 0}\frac{\cos{x}-1}{x}\\=0.$$
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