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Question
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
sin (x + a)
Solution
Let f(x) = sin (x + a)
f(x + h) = sin (x + h + a)
By first principle,
f'(x) = `lim_(h->0)(f(x + h) - f(x))/h`
= `lim_(h->0)(sin (x + h + a) - sin (x + a))/h`
= `lim_(h->0)1/h [2cos ((x + h + a + x + a)/2) sin ((x + h + a - x - a)/2)]`
= `lim_(h->0)1/h [(2 cos (2x + 2a + h)/2) sin (h/2)]`
= `lim_(h->0)1/h [( cos (2x + 2a + h)/2) {sin (h/2)/(h/2)}]`
= `lim_(h->0)1/h [((2x + 2a + h)/2) lim_(h->0){sin (h/2)/((h/2))}]` `["As" h ->0 => h/2 ->0]`
= `cos ((2x + 2a)/ 2) xx 1` `[lim_(x->0) (sin x)/x = 1]`
= cos (x + a)
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