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Ex 13.2, 11 - Find derivative: (i) sin x cos x (ii) sec x - Derivatives by formula - other trignometric

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  1. Class 11
  2. Important Question for exams Class 11
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Ex 13.2, 11 Find the derivative of the following functions: (i) sin x cos x Let f (x) = sin x cos x. Let u = sin x & v = cos x ∴ f(x) = uv So, f’(x) = (uv)’ = u’v + v’u Here, u = sin x So, u’ = cos x & v = cos x So, v’ = – sin x Now, f’(x) = (uv)’ = u’v + v’ u = cos x . cos x + ( – sin x) sin x = cos2x – sin2x = cos 2x Hence f’(x) = cos 2x Ex13.2, 11 Find the derivative of the following functions: (ii) sec x Let f (x) = sec x f(x) = 1﷮ cos﷮𝑥﷯﷯ Let u = 1 & v = cos 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 = 1 u’ = 0 & v = cos x v’ = – sin x Now, f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ = 0( cos﷮𝑥) − (− sin﷮𝑥) (1)﷯﷯﷮ 𝑐𝑜𝑠﷮2﷯𝑥﷯ = 0 + sin﷮𝑥﷯﷮ 𝑐𝑜𝑠﷮2﷯𝑥﷯ = sin﷮𝑥﷯﷮ 𝑐𝑜𝑠﷮2﷯𝑥﷯ = sin﷮𝑥﷯﷮ cos﷮𝑥﷯﷯ . 1﷮ cos﷮𝑥﷯﷯ = tan x . sec x Hence f’(x) = tan x . sec x Ex13.2, 11 Find the derivative of the following functions: (iii) 5 sec x + 4 cos x Let f (x) = 5 sec x + 4 cos x. Now, f’ (x) = ( 5 sec x + 4 cos x)’ = (5 sec x)’ + (4 cos x)’ = 5 (sec x . tan x) + 4 ( – sin x) = 5 sec x . tan x – 4 sin x Ex 13.2, 11 Find the derivative of the following functions: (iv) cosec x Let f (x) = cosec x f(x) = 1﷮ sin﷮𝑥﷯﷯ Let u = 1 & v = sin x ∴ f(x) = 𝑢﷮𝑣﷯ So, f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ Using quotient rule f’(x) = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ Finding u’ & v’ u = 1 u’ = 0 & v = sin x v’ = cos x Now, f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ = 0 ( sin﷮𝑥) − cos﷮𝑥 (1)﷯﷯﷮ 𝑠𝑖𝑛﷮2﷯𝑥﷯ = 0 − 𝑐𝑜𝑠 𝑥﷮ 𝑠𝑖𝑛﷮2﷯𝑥﷯ = − 𝑐𝑜𝑠 𝑥﷮ 𝑠𝑖𝑛﷮2﷯𝑥﷯ = − 𝑐𝑜𝑠 𝑥﷮ sin﷮𝑥﷯﷯ . 1﷮ sin﷮𝑥﷯﷯ = – cot x cosec x = – cosec x cot x Hence f’(x) = – cosec x cot x Ex13.2, 11 Find the derivative of the following functions: (v) f (x) = 3cot x + 5cosec x. Given f (x) = 3cot x + 5cosec x Now, f’(x) = (3cot x + 5cosec x)’ = 3(cot x)’ + 5( cosec x)’ = – 3cosec2x – 5 cot x cosec x = – 3 cosec2x – 5 cosec x cot x Ex13.2, 11 Find the derivative of the following functions: (vi) f (x) = 5sin x – 6 cos x + 7. Let f (x) = 5sin x – 6cos x + 7 f’(x) = (5 sin x – 6 cos x + 7)’ = (5 sin x)’ – (6 cos x)’ + (7)’ = 5 cos x – 6 ( – sin x ) + 0 = 5 cos x + 6 sin x Ex13.2, 11 Find the derivative of the following functions: (vii) f (x) = 2 tan x – 7 sec x f (x) = 2 tan x – 7 sec x. Now, f’(x) = (2 tan x – 7 sec x)’ = (2tan x)’ – (7sec x)’ = 2 (tan x)’ – 7 (sec x)’ = 2 sec2 x – 7 (sec x tan x) = 2 sec2 x – 7 sec x tan x

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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