# Example 22 - Chapter 13 Class 11 Limits and Derivatives

Last updated at March 9, 2017 by Teachoo

Last updated at March 9, 2017 by Teachoo

Transcript

Example 22 Find the derivative of (i) x5 − cosx sinx Let f(x) = x5− cosx sinx Let u = x5 – cos x & v = sin x So, f(x) = 𝑢𝑣 ∴ f’(x) = 𝑢𝑣′ Using quotient rule f’(x) = 𝑢′𝑣 − 𝑣′𝑢 𝑣2 Finding u’ & v’ u = x5 – cos x u’ = 5. x5 – 1 – ( – sin x) = 5x4 + sin x v = sin x v’ = cos x Now, f’(x) = 𝑢𝑣′ = 𝑢′𝑣 − 𝑣′𝑢 𝑣2 = (5x4 + sinx) sin x − cos x(x5 − cos x) sin2x = 5x4 sinx + sin 2x − cos x . x5 + cos2 xsin2 x = −x5 cosx + 5x4 sinx + 𝐬𝐢𝐧𝟐 𝐱 + 𝐜𝐨𝐬𝟐 𝐱 sinx2 = −x5 cosx + 5x4 sinx + 𝟏 sinx2 Thus, f’(x) = −𝐱𝟓 𝐜𝐨𝐬𝐱 + 𝟓𝐱𝟒 𝐬𝐢𝐧𝒙 + 𝟏 𝐬𝐢𝐧𝐱𝟐 Example 22 Find the derivative of (ii) 𝑥 + 𝑐𝑜𝑠𝑥 𝑡𝑎𝑛𝑥 Let f(x) = 𝑥 + 𝑐𝑜𝑠𝑥 𝑡𝑎𝑛𝑥 Let u = x + cos x & v = tan x ∴ f(x) = 𝑢𝑣 So, f’(x) = 𝑢𝑣′ Using quotient rule f’(x) = 𝑢′𝑣 − 𝑣′𝑢 𝑣2 Finding u’ & v’ u = x + cos x u’ = (x + cos x)’ = 1 – sin x v = tan x v’ = sec2x Now, f’(x) = 𝑢𝑣′ = 𝑢′𝑣 − 𝑣′𝑢 𝑣2 = (𝟏− 𝐬𝐢𝐧𝒙) ( 𝐭𝐚𝐧𝒙) − 𝒔𝒆𝒄𝟐𝒙 (𝒙 + 𝐜𝐨𝐬𝒙) ( 𝐭𝐚𝐧𝒙)𝟐

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.