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> sin'(0) = 1.

= 0, not 1, unless I'm misunderstanding your notation (or blowing trig).

The ' stands for (first-order) derivative. The second would be '' and so on.

sin(0)=0, sin'(0)=1

Ah, thanks! I was thinking it was inverse (or typo); didn’t get to d sin(x)/x.