Last updated at Sept. 17, 2019 by Teachoo

Transcript

Misc 30 Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 𝑥𝑠𝑖𝑛𝑛 𝑥 Let f(x) = 𝑥𝑠𝑖𝑛𝑛 𝑥 Let u = x & v = sinn x ∴ f(x) = 𝑢𝑣 So, f’(x) = 𝑢𝑣′ Using quotient rule f’(x) = 𝑢′𝑣 − 𝑣′𝑢 𝑣2 Finding u’ & v’ u = x u’ = (x)’ = 1 v = sinn x Let p = sin x v = pn By Leibnitz product rule v’ = (pn)’ p’ = n pn – 1 p’ Putting p = sin x = n sinn – 1 x (sin x)’ = n sinn – 1 x cos x Now, f’(x) = 𝑢𝑣′ = 𝑢′𝑣 − 𝑣′𝑢 𝑣2 = 1 sin𝑛 𝑥 − 𝑛 𝑠𝑖𝑛𝑛−1𝑥 cos𝑥 (𝑥) (𝑠𝑖𝑛𝑛𝑥)2 = 𝑠𝑖𝑛𝑛𝑥 − 𝑥 (𝑛 𝑠𝑖𝑛𝑛−1𝑥 cos𝑥) (𝑠𝑖𝑛𝑛𝑥)2 = 𝑠𝑖𝑛𝑛−1𝑥 . sin𝑥 − 𝑥 (𝑛 𝑠𝑖𝑛𝑛−1𝑥 cos𝑥) (𝑠𝑖𝑛𝑛𝑥)2 = 𝑠𝑖𝑛𝑛−1𝑥 (sin𝑥 − 𝑛𝑥 . cos𝑥) 𝑠𝑖𝑛2𝑛𝑥 = sin𝑥 − 𝑛𝑥 cos𝑥 𝑠𝑖𝑛2𝑛𝑥 . 𝑠𝑖𝑛− 𝑛−1𝑥 = sin𝑥 − 𝑛𝑥 cos𝑥 𝑠𝑖𝑛(2𝑛 − 𝑛+1)𝑥 = sin𝑥 − 𝑛𝑥 cos𝑥 𝑠𝑖𝑛𝑛 + 1𝑥 Thus, f’(x) = 𝒔𝒊𝒏𝒙 − 𝒏𝒙 𝒄𝒐𝒔𝒙 𝒔𝒊𝒏𝒏 + 𝟏𝒙

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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.