Example 22 (i) - Chapter 12 Class 11 Limits and Derivatives

Last updated at Dec. 16, 2024 by

Example 22 - Find derivative of (i) x5 - cos x / sin x - Examples

Example 22 - Chapter 13 Class 11 Limits and Derivatives - Part 2
Example 22 - Chapter 13 Class 11 Limits and Derivatives - Part 3

 

 

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Example 22 Find the derivative of (i) (x^5 − cos⁡x)/sin⁡x Let f(x) = (x^5 − cos⁡x)/sin⁡x 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) 〗)/sin2⁡x = (5x4 sin⁡〖x + sin 2⁡x − cos x . x5 + cos2 x〗)/(sin2 x) = (−x5 cos⁡〖x + 5x4 sin⁡x + 𝐬𝐢𝐧𝟐 𝐱 + 𝐜𝐨𝐬𝟐 𝐱〗)/(sin⁡x )2 = (−x5 cos⁡〖x + 5x4 sin⁡x + 𝟏〗)/(sin⁡x )2 Thus, f’(x) = (−𝐱𝟓 𝐜𝐨𝐬⁡〖𝐱 + 𝟓𝐱𝟒 𝐬𝐢𝐧⁡𝒙 + 𝟏〗)/(𝐬𝐢𝐧⁡𝐱 )𝟐 (Using sin2x + cos2x = 1)

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo