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  1. Chapter 13 Class 11 Limits and Derivatives
  2. Serial order wise

Transcript

Example 21 Compute derivative of (i) f(x) = sin 2x Let f (x) = sin 2x = 2 sin x cos x Let u = 2 sin x & v = cos x So, f(x) = uv ∴ f’(x) = (uv)’ = u’v + v’u Here, u = 2 sin x u’ = 2 cos x & v = cos x v’ = – sin x f’(x) = (uv)’ = u’v + v’ u = 2 cos x . cos x + 2 sin x ( – sin x) = 2 cos2 x – 2 sin2 x = 2 (cos2 x – sin2 x) ∴ f’(x) = 2 (cos2 x – sin2 x) (𝐷𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑠𝑖𝑛⁡〖𝑥=𝑐𝑜𝑠⁡𝑥 〗) (𝐷𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑐𝑜𝑠⁡〖𝑥=〖− 𝑠𝑖𝑛〗⁡𝑥 〗) Example 21 Compute derivative of (ii) g(x) = cot x g(x) = cot x = cos⁡𝑥/sin⁡𝑥 Let u = cos x & v = sin x ∴ g(x) = 𝑢/𝑣 So, g’(x) = (𝑢/𝑣)^′ Using quotient rule g’(x) = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 Finding u’ & v’ u = cos x u’ = – sin x & v = sin x v’ = cos x Now, f’(x) = (𝑢/𝑣)^′ = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 (𝐷𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑐𝑜𝑠⁡〖𝑥=〖− 𝑠𝑖𝑛 〗⁡𝑥 〗 ) (𝐷𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑠𝑖𝑛⁡〖𝑥=〖𝑐𝑜𝑠 〗⁡𝑥 〗 ) = (−sin⁡〖𝑥 (sin⁡𝑥 ) −〖 cos〗⁡〖𝑥 (cos⁡〖𝑥)〗 〗 〗)/(〖𝑠𝑖𝑛〗^2 𝑥) = (−sin2⁡〖𝑥 −〖 cos2〗⁡〖𝑥 〗 〗)/(〖𝑠𝑖𝑛〗^2 𝑥) = (−(𝐬𝐢𝐧𝟐⁡〖𝒙 + 〖 𝐜𝐨𝐬𝟐〗⁡〖𝒙) 〗 〗)/(〖𝑠𝑖𝑛〗^2 𝑥) = (−𝟏)/(〖𝑠𝑖𝑛〗^2 𝑥) = –cosec2x Hence, f’(x) = –cosec2x (𝑈𝑠𝑖𝑛𝑔 𝑠𝑖𝑛2𝑥+𝑐𝑜𝑠2𝑥=1)

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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.