f(x) = sinx. Using differentiation from first principles find the exact value of f' (π/6).
The derivative of the function where x=π/6 is defined asThe limit as h->0 of [sin(h+π/6)-sin(π/6)]/hUsing the double angle formula, sin(h+π/6) = sin(h)cos(π/6) + cos(h)sin(π/6) = √3sin(h)/2 + cos(h)sin(π/6)The limit becomes [sin(h)/2 + cos(h)sin(π/6)-sin(π/6)]/hThe limit can be broken up into two partslim as h->0 of [cos(h)sin(π/6)-sin(π/6)]/h = 0 (could use l'Hospital's rule or half angle formula)lim as h->0 of [√3sin(h)/2]/h = 1/2 (small angle approximation)0+√3/2=√3/2
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