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Last updated at March 22, 2023 by Teachoo

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) = (ππ₯ + π)/(ππ₯2 + ππ₯ + π) Let u = ax + b & v = px2 +qx+ r β΄ f(x) = π’/π£ So, fβ(x) = (π’/π£)^β² fβ(x) = (π’^β² π£ β π£^β² π’)/π£^2 Finding uβ & vβ u = ax + b uβ = a Γ 1 + 0 uβ = a v = px2 + qx + r vβ= p Γ 2x + q Γ 1 + 0 vβ = 2px + q fβ(x) = (π’/π£)^β² = (π’^β² π£ βγ π£γ^β² π’)/π£^2 = (π (ππ₯2 + ππ₯ + π ) β (2ππ₯ + π) (ππ₯ + π) )/(ππ₯2+ ππ₯ + π)2 = (πππ₯2 + πππ₯ + ππ β 2ππ₯ (ππ₯ + π) β π (ππ₯ + π) )/(ππ₯2+ ππ₯ + π)2 = (πππ₯2 β 2πππ₯2 + πππ₯ β πππ₯ β 2πππ₯ β ππ + ππ )/(ππ₯2+ ππ₯ + π)2 = (β πππ₯2 β 2πππ₯ β ππ + ππ )/(ππ₯2+ ππ₯ + π)2 = (β πππ₯2 β 2πππ₯ + ππ β ππ)/(ππ₯2+ ππ₯ + π)2 So, fβ(x) = (β ππππ β ππππ + ππ β ππ)/(πππ+ ππ + π)π