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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))/ cos x`
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Solution
Let f(x) = `(sin (x + a))/(cos x)`
By quotient rule,
f'(x) = `(cos x d/dx [sin (x + a)] - sin(x + a) d/dx cos x)/cos^2 x`
f'(x) = `(cos x d/dx [sin (x + a)] - sin(x + a) (-sin x))/cos^2 x` ...(i)
Let g(x) = sin (x + a) Accordingly. g(x + h) = sin (x + h + a)
By first principle,
g'(x) = `lim_(h->0) (g(x + h) - g(x))/h`
= `lim_(h->0)1/h [sin (x + h + a) -sin (x + a)]`
= `lim_(h->0)1/h [2 cos ((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) [cos ((2x + 2a + h)/2) {sin (h/2)/(h/2)}]`
= `lim_(h->0) cos ((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_(h->0) (sin h)/h = 1]`
= cos (x + a)
From (i) and (ii) we obtain
f'(x) = `(cosx. cos (x + a) + sin x sin (x + a))/cos^2x`
= `(cos (x + a - x))/cos^2 x`
= `(cos a)/cos^2 x`
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