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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) = (π‘Žπ‘₯ + 𝑏)/(𝑝π‘₯2 + π‘žπ‘₯ + π‘Ÿ) Let u = ax + b & v = px2 +qx+ r ∴ f(x) = 𝑒/𝑣 So, f’(x) = (𝑒/𝑣)^β€² f’(x) = (𝑒^β€² 𝑣 βˆ’ 𝑣^β€² 𝑒)/𝑣^2 Finding u’ & v’ u = ax + b u’ = a Γ— 1 + 0 u’ = a v = px2 + qx + r v’= p Γ— 2x + q Γ— 1 + 0 v’ = 2px + q f’(x) = (𝑒/𝑣)^β€² = (𝑒^β€² 𝑣 βˆ’γ€– 𝑣〗^β€² 𝑒)/𝑣^2 = (π‘Ž (𝑝π‘₯2 + π‘žπ‘₯ + π‘Ÿ ) βˆ’ (2𝑝π‘₯ + π‘ž) (π‘Žπ‘₯ + 𝑏) )/(𝑝π‘₯2+ π‘žπ‘₯ + π‘Ÿ)2 = (π‘Žπ‘π‘₯2 + π‘Žπ‘žπ‘₯ + π‘Žπ‘Ÿ βˆ’ 2𝑝π‘₯ (π‘Žπ‘₯ + 𝑏) βˆ’ π‘ž (π‘Žπ‘₯ + 𝑏) )/(𝑝π‘₯2+ π‘žπ‘₯ + π‘Ÿ)2 = (π‘Žπ‘π‘₯2 βˆ’ 2π‘Žπ‘π‘₯2 + π‘Žπ‘žπ‘₯ βˆ’ π‘Žπ‘žπ‘₯ βˆ’ 2𝑝𝑏π‘₯ βˆ’ π‘žπ‘ + π‘Žπ‘Ÿ )/(𝑝π‘₯2+ π‘žπ‘₯ + π‘Ÿ)2 = (βˆ’ π‘Žπ‘π‘₯2 βˆ’ 2π‘Žπ‘π‘₯ βˆ’ π‘žπ‘ + π‘Žπ‘Ÿ )/(𝑝π‘₯2+ π‘žπ‘₯ + π‘Ÿ)2 = (βˆ’ π‘Žπ‘π‘₯2 βˆ’ 2π‘Žπ‘π‘₯ + π‘Žπ‘Ÿ βˆ’ π‘π‘ž)/(𝑝π‘₯2+ π‘žπ‘₯ + π‘Ÿ)2 So, f’(x) = (βˆ’ π’‚π’‘π’™πŸ βˆ’ πŸπ’‚π’ƒπ’™ + 𝒂𝒓 βˆ’ 𝒃𝒒)/(π’‘π’™πŸ+ 𝒒𝒙 + 𝒓)𝟐

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