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Last updated at March 22, 2023 by Teachoo

Misc 18 Find the derivative of the following functions (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〗 Let f (x) = sec〖x − 1〗/sec〖x + 1〗 Let u = sec x – 1 & v = sec x + 1 ∴ f(x) = 𝑢/𝑣 So, f’(x) = (𝑢/𝑣)^′ Using quotient rule f’(x) = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 Finding u’ & v’ u = sec x – 1 u’ = (sec x – 1)’ = sec x tan x – 0 = sec x tan x & v = sec x + 1 v’= sec x tan x + 0 = sec x tan x Now, f’(x) = (𝑢/𝑣)^′ Derivative of sec x = sec x tan x Derivative of constant = 0 = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 = ( (sec〖𝑥 tan〖𝑥)〗 (sec〖𝑥 + 1) − (sec〖𝑥 tan〖𝑥)〗 (sec〖𝑥 − 1)〗 〗 〗 〗)/〖(sec〖x + 1〗)〗^2 = ( sec〖𝑥 . tan𝑥 [(sec〖𝑥 + 1) − (sec〖𝑥 − 1)] 〗 〗 〗)/〖(sec〖x + 1〗)〗^2 = ( sec〖𝑥 . tan𝑥 (sec〖𝑥 + 1−〖 sec〗〖𝑥 + 1〗) 〗 〗)/〖(sec〖x + 1〗)〗^2 = sec〖𝑥 tan〖𝑥 (2 + 0)〗 〗/〖(sec〖𝑥 + 1〗)〗^2 = (𝟐 𝐬𝐞𝐜〖𝒙 𝐭𝐚𝐧𝒙 〗)/〖(𝒔𝒆𝒄〖𝒙 + 𝟏〗)〗^𝟐