Last updated at Sept. 6, 2021 by Teachoo
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Misc 12 (Method 1) 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)n Let f(x) = (ax + b)n. We know that f’(x) = limh→0 f x + h − f(x)h Here, f(x) = (ax + b)n So, f(x + h) = (a(x + h) + b)n Putting values f’(x) = limh→0 𝑎 𝑥+ℎ+𝑏𝑛 − 𝑎𝑥 + 𝑏𝑛h f’(x) = limh→0 𝑎 𝑥+ℎ+𝑏𝑛 − 𝑎𝑥 + 𝑏𝑛h = limh→0 𝑎𝑥 + 𝑎ℎ + 𝑏𝑛 − 𝑎𝑥 + 𝑏𝑛h = limh→0 𝑎𝑥 + 𝑏 + 𝑎ℎ𝑛 − 𝑎𝑥 + 𝑏𝑛h = limh→0 𝑎𝑥 + 𝑏 1+ 𝑎ℎ𝑎𝑥 + 𝑏𝑛 − 𝑎𝑥 + 𝑏𝑛h = limh→0 𝑎𝑥 + 𝑏𝑛 𝟏+ 𝒂𝒉𝒂𝒙 + 𝒃𝒏− 1h = limh→0(ax + b)n 𝟏𝒏 + 𝒏𝑪𝟏 𝒂𝒏−𝟏 𝒂𝒉𝒂𝒙 + 𝒃𝟏+ 𝒏𝑪𝟐 𝟏𝒏−𝟐 𝒂𝒉𝒂𝒙 + 𝒃𝟐+…. + 𝒂𝒉𝒂𝒙 + 𝒃𝒏 − 1h = limh→0(ax + b)n 1+ 𝑛!1! 𝑛 − 1! 𝑎ℎ𝑎𝑥 + 𝑏+ 𝑛𝐶2 𝑎ℎ𝑎𝑥 + 𝑏2+…. + 𝑎ℎ𝑎𝑥 + 𝑏𝑛−1h = limh→0(ax + b)n 1 + 𝑛 . 𝑎ℎ𝑎𝑥 + 𝑏+ 𝑛𝐶2 𝑎ℎ𝑎𝑥 + 𝑏2+…. + 𝑎ℎ𝑎𝑥 + 𝑏𝑛−1h = limh→0(ax + b)n 1 − 1 + 𝑛 𝑎ℎ𝑎𝑥 + 𝑏+ 𝑛𝐶2 𝑎ℎ𝑎𝑥 + 𝑏2+…. + 𝑎ℎ𝑎𝑥 + 𝑏𝑛h = limh→0(ax + b)n 0 + 𝑛 𝑎ℎ𝑎𝑥 + 𝑏+ 𝑛𝐶2 𝑎ℎ𝑎𝑥 + 𝑏2+…. + 𝑎ℎ𝑎𝑥 + 𝑏𝑛h = limh→0(ax + b)n 𝑎ℎ𝑛𝑎𝑥 + 𝑏+ 𝑛𝐶2 𝑎ℎ𝑎𝑥 + 𝑏2+…. + 𝑎ℎ𝑎𝑥 + 𝑏𝑛h = limh→0(ax + b)n ℎ 𝑎𝑛𝑎𝑥 + 𝑏+ 𝑛𝐶2 ℎ2 𝑎𝑎𝑥 + 𝑏2+…. + ℎ𝑛 𝑎𝑎𝑥 + 𝑏𝑛h = limh→0(ax + b)n ℎ 𝑎𝑛𝑎𝑥 + 𝑏+ 𝑛𝐶2 ℎ 𝑎𝑎𝑥 + 𝑏2+…. + ℎ𝑛−1 𝑎𝑎𝑥 + 𝑏𝑛h = limh→0(ax + b)n 𝑎𝑛𝑎𝑥 + 𝑏+ 𝑛𝐶2 ℎ 𝑎𝑎𝑥 + 𝑏2+…. + ℎ𝑛−1 𝑎𝑎𝑥 + 𝑏𝑛 Putting h = 0 = limh→0(ax + b)n 𝑎𝑛𝑎𝑥 + 𝑏+ 𝑛𝐶2 (0) 𝑎𝑎𝑥 + 𝑏2+…. + (0)𝑛−1 𝑎𝑎𝑥 + 𝑏𝑛 = (ax + b)n 𝑎𝑛𝑎𝑥 + 𝑏+0+…+0 = (ax + b)n . 𝑎𝑛(𝑎𝑥 + 𝑏) = an (ax + b)n – 1 Hence f’ (x) = an (ax + b)n – 1 Misc12 (Method 2) 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)n Let f(x) = (ax + b)n. Let p = (ax + b) So, f(x) = pn Now, f’(x) = (pn)’ f’(x) = npn–1 p’ Putting p = (ax + b) f’(x) = n (ax + b)n – 1 . (ax + b)’ f’(x) = n (ax + b)n – 1 . (ax + b)’ = n (ax + b)n – 1 [a . 1 . x1 – 1 + 0] = n (ax + b)n – 1 [a . 1 . x1 – 1 + 0] = n ( ax + b) n – 1 (ax0) = n (ax + b)n – 1 (a) = an (ax + b)n – 1
Miscellaneous (Term 1 and Term 2)
Misc 1 (ii) Important
Misc 1 (iii)
Misc 1 (iv) Important
Misc 2
Misc 3 Important
Misc 4 Important
Misc 5
Misc 6 Important
Misc 7
Misc 8 Important
Misc 9 Important
Misc 10
Misc 11
Misc 12 Important You are here
Misc 13
Misc 14 Important
Misc 15
Misc 16
Misc 17 Important
Misc 18 Important
Misc 19
Misc 20 Important
Misc 21
Misc 22 Important
Misc 23
Misc 24 Important
Misc 25
Misc 26
Misc 27 Important
Misc 28 Important
Misc 29 Important
Misc 30 Important
Miscellaneous (Term 1 and Term 2)
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