Misc 8 - Find derivative: ax + b / px2 + qx + r - Class 11

Misc 8 - Chapter 13 Class 11 Limits and Derivatives - Part 2
Misc 8 - Chapter 13 Class 11 Limits and Derivatives - Part 3

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Misc 8 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): (ax + b)/(px2 + qx + r) Let f(x) = (𝑎𝑥 + 𝑏)/(𝑝𝑥2 + 𝑞𝑥 + 𝑟) Let u = ax + b & v = px2 +qx+ r ∴ f(x) = 𝑢/𝑣 So, f’(x) = (𝑢/𝑣)^′ f’(x) = (𝑢^′ 𝑣 − 𝑣^′ 𝑢)/𝑣^2 Finding u’ & v’ u = ax + b u’ = a × 1 + 0 u’ = a v = px2 + qx + r v’= p × 2x + q × 1 + 0 v’ = 2px + q f’(x) = (𝑢/𝑣)^′ = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 = (𝑎 (𝑝𝑥2 + 𝑞𝑥 + 𝑟 ) − (2𝑝𝑥 + 𝑞) (𝑎𝑥 + 𝑏) )/(𝑝𝑥2+ 𝑞𝑥 + 𝑟)2 = (𝑎𝑝𝑥2 + 𝑎𝑞𝑥 + 𝑎𝑟 − 2𝑝𝑥 (𝑎𝑥 + 𝑏) − 𝑞 (𝑎𝑥 + 𝑏) )/(𝑝𝑥2+ 𝑞𝑥 + 𝑟)2 = (𝑎𝑝𝑥2 − 2𝑎𝑝𝑥2 + 𝑎𝑞𝑥 − 𝑎𝑞𝑥 − 2𝑝𝑏𝑥 − 𝑞𝑏 + 𝑎𝑟 )/(𝑝𝑥2+ 𝑞𝑥 + 𝑟)2 = (− 𝑎𝑝𝑥2 − 2𝑎𝑏𝑥 − 𝑞𝑏 + 𝑎𝑟 )/(𝑝𝑥2+ 𝑞𝑥 + 𝑟)2 = (− 𝑎𝑝𝑥2 − 2𝑎𝑏𝑥 + 𝑎𝑟 − 𝑏𝑞)/(𝑝𝑥2+ 𝑞𝑥 + 𝑟)2 So, f’(x) = (− 𝒂𝒑𝒙𝟐 − 𝟐𝒂𝒃𝒙 + 𝒂𝒓 − 𝒃𝒒)/(𝒑𝒙𝟐+ 𝒒𝒙 + 𝒓)𝟐

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.