Misc 9 - Find derivative: px2 + qx + r / ax + b - Chapter 13 - Miscellaneous

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Misc 9 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): px2 + qx + r﷮ax + b﷯ Let f(x) = px2 + qx + r﷮ax + b﷯ Let u = px2 + qx + r & v = ax + b ∴ f(x) = 𝑢﷮𝑣﷯ So, f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ f’(x) = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ Finding u’ & v’ u = px2 + qx + r u’ = p . 2 .x2 – 1 + q . 1 + 0 u’ = 2px + q v = ax + 3 v’= a. 1 + 0 v’ = a f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ = 2𝑝𝑥 + 𝑞﷯ 𝑎𝑥 + 𝑏﷯ − a px2 + 𝑞𝑥 + 𝑟﷯ ﷮ ax + b﷯2﷯ = 2𝑎𝑝𝑥2 + 2𝑏𝑝𝑥 + 𝑞𝑎𝑥 + 𝑏𝑞 − 𝑎𝑝𝑥2 − 𝑞𝑎𝑥 − 𝑎𝑟﷮ ax + b﷯2﷯ = 𝑎𝑝𝑥2 + 2𝑏𝑝𝑥 + 𝑏𝑞 − 𝑎𝑟 + 0﷮ ax + b﷯2﷯ ∴ f’(x) = 𝒂𝒑𝒙𝟐 + 𝟐𝒃𝒑𝒙 + 𝒃𝒒 − 𝒂𝒓﷮ 𝐚𝐱 + 𝒃﷯𝟐﷯

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