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

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): 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’ = 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) = (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)

Miscellaneous 