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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):
`(sec x - 1)/(sec x + 1)`
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Solution
Let f(x) = `(sec x - 1)/(sec x + 1)`
f(x) = `(1/cos x -1)/(1/cos x +1)`
= `(1 - cos x)/(1 + cos x)`
By quotient rule,
f'(x) = `((1 + cosx)d/dx(1 - cosx)-(1 - cos x)d/dx(1 + cos x))/((1 + cos x^2))`
= `((1 + cos x) (sin x) - (1 - cos x) (-sin x))/((1 + cos x)^2)`
= `(sin x + cos x sin x + sin x - sin x cos x) /(1 + cos x)^2`
= `(2 sin x)/(1 + cos x)^2`
= `(2 sin x)/(1 + 1/sec x)^2 = (2 sin x)/((sec x + 1)^2/(sec^2 x))`
= `(2 sin x sec^2x)/ (secx+1)^2`
= `((2 sin x)/(cos x)sec x)/(sec x + 1)^2`
= `(2sec x tan x)/(sec x + 1)^2`
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