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Question
Find the derivative of f(x) = sin x, by first principle.
Solution
By definition,
f'(x) = `lim_(h -> 0) (f(x + h) - f(x))/h`
= `lim_(h -> 0) (sin(x + h) - sinx)/h`
= `lim_(h -> 0) (2cos (2x + h)/2 sin h/2)/(2 * h/2)`
= `lim_(h -> 0) cos ((2x + h))/2 * lim_(h -> 0) (sin h/2)/(h/2)`
= `cos x . 1`
= cos x
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