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Misc 30 - Find derivative: x / sinn x - Chapter 13 Class 11 - Derivatives by formula - sin & cos

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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): 𝑥﷮𝑠𝑖𝑛𝑛 𝑥﷯ Let f(x) = 𝑥﷮𝑠𝑖𝑛𝑛 𝑥﷯ Let u = x & v = sinn 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 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) = 𝑢﷮𝑣﷯﷯﷮′﷯ = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ = 1 sin𝑛﷮ 𝑥﷯﷯ − 𝑛 𝑠𝑖𝑛﷮𝑛−1﷯𝑥 cos﷮𝑥 (𝑥)﷯﷮ (𝑠𝑖𝑛﷮𝑛﷯𝑥)﷮2﷯﷯ = 𝑠𝑖𝑛﷮𝑛﷯𝑥 − 𝑥 (𝑛 𝑠𝑖𝑛﷮𝑛−1﷯𝑥 cos﷮𝑥) ﷯﷮ (𝑠𝑖𝑛﷮𝑛﷯𝑥)﷮2﷯﷯ = 𝑠𝑖𝑛﷮𝑛−1﷯𝑥 . sin﷮𝑥 − 𝑥 (𝑛 ﷯ 𝑠𝑖𝑛﷮𝑛−1﷯𝑥 cos﷮𝑥) ﷯﷮ (𝑠𝑖𝑛﷮𝑛﷯𝑥)﷮2﷯﷯ = 𝑠𝑖𝑛﷮𝑛−1﷯𝑥 (sin﷮𝑥 − 𝑛𝑥 . ﷯ cos﷮𝑥) ﷯﷮ 𝑠𝑖𝑛﷮2𝑛﷯𝑥﷯ = sin﷮𝑥 − 𝑛𝑥 cos﷮𝑥﷯﷯﷮ 𝑠𝑖𝑛﷮2𝑛﷯𝑥 . 𝑠𝑖𝑛﷮− 𝑛−1﷯﷯𝑥﷯ = sin﷮𝑥 − 𝑛𝑥 cos﷮𝑥﷯﷯﷮ 𝑠𝑖𝑛﷮(2𝑛 − 𝑛+1)﷯𝑥﷯ = sin﷮𝑥 − 𝑛𝑥 cos﷮𝑥﷯﷯﷮ 𝑠𝑖𝑛﷮𝑛 + 1﷯𝑥﷯ Thus, f’(x) = 𝒔𝒊𝒏﷮𝒙 − 𝒏𝒙 𝒄𝒐𝒔﷮𝒙﷯﷯﷮ 𝒔𝒊𝒏﷮𝒏 + 𝟏﷯𝒙﷯

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