In other words, f'(x) drops to zero when , and becomes -1 by the time x reaches pi (see the picture). Therefore, f'(x) is a function which starts at 1 when x=0, decreases to 0 when , drops to -1 when , rises back to 0 when , and so on. This is precisely what the cosine function does, so it should be no surprise that f'(x) = cos x. Similar reasoning shows why it is reasonable that, when f(x)=cos x, f'(x)=-sin x.
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