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

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): 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
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.