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Misc 18 - Find derivative: sec x - 1 / sec x + 1 - CBSE - Derivatives by formula - sin & cos

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  1. Chapter 13 Class 11 Limits and Derivatives
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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 f’(x) = 𝑢﷮𝑣﷯﷯′ = 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 . tan﷮𝑥﷯ [( sec﷮𝑥 + 1) − ( sec﷮𝑥 − 1)] ﷯﷯﷯﷮ ( sec﷮x + 1﷯)﷮2﷯﷯ = sec﷮𝑥 . tan﷮𝑥﷯ ( sec﷮𝑥 + 1− sec﷮𝑥 + 1﷯) ﷯﷯﷮ ( sec﷮x + 1﷯)﷮2﷯﷯ = sec﷮𝑥 . tan﷮𝑥﷯ (1 + 1 + sec﷮𝑥 − sec﷮𝑥)﷯﷯﷯﷮ ( sec﷮x + 1﷯)﷮2﷯﷯ = sec﷮𝑥 tan﷮𝑥 (2 + 0)﷯﷯﷮ ( sec﷮𝑥 + 1﷯)﷮2﷯﷯ = 2 sec﷮𝑥 tan﷮𝑥﷯﷯﷮ ( sec﷮𝑥 + 1﷯)﷮2﷯﷯ Hence f’(x) = 𝟐 𝒔𝒆𝒄﷮𝒙 𝒕𝒂𝒏﷮𝒙﷯﷯﷮ ( 𝒔𝒆𝒄﷮𝒙 + 𝟏﷯)﷮𝟐﷯﷯

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