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Misc 30 - Find derivative: x / sinn x - Chapter 13 Class 11

Misc 30 - Chapter 13 Class 11 Limits and Derivatives - Part 2
Misc 30 - Chapter 13 Class 11 Limits and Derivatives - Part 3
Misc 30 - Chapter 13 Class 11 Limits and Derivatives - Part 4

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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): π‘₯/(𝑠𝑖𝑛𝑛 π‘₯) Let f(x) = π‘₯/(𝑠𝑖𝑛𝑛 π‘₯) Let u = x & v = sinn x ∴ f(x) = 𝑒/𝑣 So, f’(x) = (𝑒/𝑣)^β€² Using quotient rule f’(x) = (𝑒^β€² 𝑣 βˆ’γ€– 𝑣〗^β€² 𝑒)/𝑣^2 Finding u’ & v’ u = x u’ = 1 Now, 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) = (𝑒/𝑣)^β€² = (𝑒^β€² 𝑣 βˆ’γ€– 𝑣〗^β€² 𝑒)/𝑣^2 = ( 1 (sin𝑛⁑〖 π‘₯γ€— ) βˆ’ 〖𝑛 𝑠𝑖𝑛〗^(π‘›βˆ’1) π‘₯ cos⁑〖π‘₯ (π‘₯)γ€—)/γ€–γ€–(𝑠𝑖𝑛〗^𝑛 π‘₯)γ€—^2 = ( 〖𝑠𝑖𝑛〗^𝑛 π‘₯ βˆ’ π‘₯ (𝑛〖𝑠𝑖𝑛〗^(π‘›βˆ’1) π‘₯ cos⁑〖π‘₯) γ€—)/γ€–γ€–(𝑠𝑖𝑛〗^𝑛 π‘₯)γ€—^2 = ( γ€–π’”π’Šπ’γ€—^(π’βˆ’πŸ) 𝒙 . sin⁑〖π‘₯ βˆ’ π‘₯ (𝑛 γ€— 〖𝑠𝑖𝑛〗^(π‘›βˆ’1) π‘₯ cos⁑〖π‘₯) γ€—)/γ€–γ€–(𝑠𝑖𝑛〗^𝑛 π‘₯)γ€—^2 = ( γ€–π’”π’Šπ’γ€—^(π’βˆ’πŸ) 𝒙 γ€–(sin〗⁑〖π‘₯ βˆ’ 𝑛π‘₯ . γ€— cos⁑〖π‘₯) γ€—)/(〖𝑠𝑖𝑛〗^2𝑛 π‘₯) = sin⁑〖π‘₯ βˆ’ 𝑛π‘₯ cos⁑π‘₯ γ€—/(〖𝑠𝑖𝑛〗^2𝑛 𝒙 . γ€–π’”π’Šπ’γ€—^(βˆ’(π’βˆ’πŸ) ) 𝒙) = sin⁑〖π‘₯ βˆ’ 𝑛π‘₯ cos⁑π‘₯ γ€—/(γ€–π’”π’Šπ’γ€—^((πŸπ’ βˆ’ 𝒏+𝟏)) 𝒙) = sin⁑〖π‘₯ βˆ’ 𝑛π‘₯ cos⁑π‘₯ γ€—/(〖𝑠𝑖𝑛〗^(𝑛 + 1) π‘₯) Thus, f’(x) = π’”π’Šπ’β‘γ€–π’™ βˆ’ 𝒏𝒙 𝒄𝒐𝒔⁑𝒙 γ€—/(γ€–π’”π’Šπ’γ€—^(𝒏 + 𝟏) 𝒙)

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.