Misc 25 - Find derivative: (x + cos x) (x - tan x) - Derivatives by formula - other trignometric

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  1. Chapter 13 Class 11 Limits and Derivatives
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Misc 25 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 + cos x) (x – tan x) Let f (x) = (x + cos x) (x – tan x) Let u = x + cos x & v = x – tan x ∴ f(x) = uv So, f’(x) = 𝑢𝑣﷯﷮′﷯ Using product rule f’(x) = 𝑢﷮′﷯𝑣+ 𝑣﷮′﷯𝑢 Finding u’ & v’ u = x + cos x u’ = (x + cos x)’ = 1 – sin x & v = x – tan x v’ = (x – tan x)’ = 1 – sec2 x Now, f’(x) = 𝑢𝑣﷯′ = 𝑢﷮′﷯𝑣+ 𝑣﷮′﷯𝑢 = (1 – sin x) (x – tan x) + (1 – sec2 x) (x + cos x) = (1 – sin x) (x – tan x) – (sec2 x – 1) (x +cos x) = (1 – sin x) (x – tan x) – tan2x (x + cos x) = – tan2x (x + cos x) + (x – tan x) (1 – sin x)

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