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Misc 28 - Find derivative: x / 1 + tan x - Chapter 13 Class 11 - Derivatives by formula - other trignometric

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Misc 28 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): 𝑥﷮1 + 𝑡𝑎𝑛﷮𝑥﷯ ﷯ Let f (x) = 𝑥﷮1 + 𝑡𝑎𝑛﷮𝑥﷯ ﷯ Let u = x & v = 1 + tan x So, f(x) = 𝑢﷮𝑣﷯ ∴ f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ Using quotient rule f’(x) = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ Finding u’ & v’ u = x u’ = (x)’ = 1 & v = 1 + tan x v’ = (1 + tan x)’ = 0 + sec2 x = sec2 x Now, f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ = 1(1 + tan﷮𝑥)﷯ − 𝑠𝑒𝑐2 𝑥 (𝑥)﷮ (1 + tan﷮𝑥)﷯﷮2﷯﷯ = 𝟏 + 𝒕𝒂𝒏﷮𝒙 − 𝒙 𝒔𝒆𝒄𝟐 𝒙﷯﷮ (𝟏 + 𝒕𝒂𝒏﷮𝒙)﷯﷮𝟐﷯﷯

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