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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.