# Example 22 - Chapter 13 Class 11 Limits and Derivatives

Last updated at Nov. 30, 2019 by Teachoo

Last updated at Nov. 30, 2019 by Teachoo

Transcript

Example 22 Find the derivative of (i) (x^5 − cosx)/sinx Let f(x) = (x^5 − cosx)/sinx Let u = x5 – cos x & v = sin 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 = x5 – cos x u’ = 5. x5 – 1 – ( – sin x) = 5x4 + sin x v = sin x v’ = cos x Now, f’(x) = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 Derivative of xn is nxn – 1 & Derivative of cos x = – sin x (Derivative of sin x = cos x) = ((5x4 + sin〖x) sin x −(cos x)(x5 − cos x) 〗)/sin2x = (5x4 sin〖x + sin 2x − cos x . x5 + cos2 x〗)/(sin2 x) = (−x5 cos〖x + 5x4 sinx + 𝐬𝐢𝐧𝟐 𝐱 + 𝐜𝐨𝐬𝟐 𝐱〗)/(sinx )2 = (−x5 cos〖x + 5x4 sinx + 𝟏〗)/(sinx )2 Thus, f’(x) = (−𝐱𝟓 𝐜𝐨𝐬〖𝐱 + 𝟓𝐱𝟒 𝐬𝐢𝐧𝒙 + 𝟏〗)/(𝐬𝐢𝐧𝐱 )𝟐 (Using sin2x + cos2x = 1) 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) = 𝑢/𝑣 So, f’(x) = (𝑢/𝑣)^′ Using quotient rule f’(x) = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 Finding u’ & v’ u = x + cos x u’ = (x + cos x)’ = 1 – sin x v = tan x v’ = sec2x Now, f’(x) = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 = ((𝟏 −〖 𝐬𝐢𝐧〗〖𝒙) (𝐭𝐚𝐧〖𝒙) − 𝒔𝒆𝒄𝟐𝒙 (𝒙 + 〖 𝐜𝐨𝐬〗〖𝒙)〗 〗 〗)/〖(𝐭𝐚𝐧〖𝒙)〗〗^𝟐 (xn)’ = n xn – 1 Derivative of cos x = –sin x Derivative of tan x = sec2x (Calculated in Example 17)

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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.