Misc 8

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 + 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. 2.x2 – 1 + 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﷯