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Last updated at Nov. 30, 2019 by Teachoo

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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=𝑐𝑜𝑠𝑥 〗) (𝐷𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑐𝑜𝑠〖𝑥=〖− 𝑠𝑖𝑛〗𝑥 〗) Example 21 Compute derivative of (ii) g(x) = cot x g(x) = cot x = cos𝑥/sin𝑥 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 (sin𝑥 ) −〖 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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About the Author

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.