

Last updated at May 29, 2018 by Teachoo
Transcript
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 + bpx2 + qx + r Let f(x) = ax + bpx2 + qx + r Let u = ax + b & v = px2 +qx+ r ∴ f(x) = 𝑢𝑣 So, f’(x) = 𝑢𝑣′ f’(x) = 𝑢′𝑣 − 𝑣′𝑢 𝑣2 Finding u’ & v’ u = ax + b u’ = a.1 + 0 u’ = a v = px2 + qx + r v’= p. 2.x2 – 1 + q. 1 + 0 v’ = 2px + q f’(x) = 𝑢𝑣′ = 𝑢′𝑣 − 𝑣′𝑢 𝑣2 = 𝑎 𝑝𝑥2 + 𝑞𝑥 + 𝑟 − 2px + 𝑞 𝑎𝑥 + 𝑏 px2+ qx + r2 = 𝑎𝑝𝑥2 + 𝑎𝑞𝑥 + 𝑎𝑟 − 2px ax + b − 𝑞 𝑎𝑥 + 𝑏 px2+ qx + r2 = 𝑎𝑝𝑥2 − 2𝑎px2 + aqx − aqx − 2pbx − qb + ar px2+ qx + r2 = − 𝑎𝑝𝑥2 − 2𝑎bx − qb + ar px2+ qx + r2 = − 𝑎𝑝𝑥2 − 2𝑎bx + ar − bq px2+ qx + r2 So, f’(x) = − 𝑎𝑝𝑥2 − 2𝑎bx + ar − bq px2+ qx + r2
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