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Ex 13.2, 8 - Find derivative of xn - an/x - a for constant a - Derivatives by formula - x^n formula

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  1. Chapter 13 Class 11 Limits and Derivatives
  2. Serial order wise
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Ex 13.2, 8 Find the derivative of xn − a﷮n﷯﷮x − a﷯ for some constant a. We need to find derivative of xn − an﷮x − a﷯ Let f(x) = xn − a﷮n﷯﷮x − a﷯ Let u = xn – an & v = x – a ∴ f(x) = 𝑢﷮𝑣﷯ So, f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ f’(x) = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ Calculating u’ u = xn – an Different w.r.t x u’ = (xn – an)’ u’ = (xn)’ – (an)’ u’ = nxn – 1 – 0 u’ = nxn – 1 Calculating v’ v = x – a Different w. r. t x v’ = (x – a)’ v’ = (x)’ – (a)’ v’ = 1 – 0 v’ = 1 f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ f’(x) = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ = 𝑛𝑥﷮𝑛−1﷯ 𝑥 − 𝑎﷯ − 1. (𝑥﷮𝑛﷯− 𝑎﷮𝑛﷯) ﷮ 𝑣﷮2﷯﷯ = 𝑛𝑥﷮𝑛−1﷯ . 𝑥 − 𝑎𝑛𝑥﷮𝑛−1﷯ − 𝑥﷮𝑛﷯ + 𝑎﷮𝑛﷯﷮ (𝑥 − 𝑎)﷮2﷯﷯ = 𝑛𝑥﷮𝑛−1+1﷯ − 𝑎𝑛𝑥﷮𝑛−1﷯ − 𝑥﷮𝑛﷯ + 𝑎﷮𝑛﷯﷮ (𝑥 − 𝑎)﷮2﷯﷯ = 𝑛𝑥﷮𝑛﷯ − 𝑎𝑛𝑥﷮𝑛−1﷯ − 𝑥﷮𝑛﷯+ 𝑎﷮𝑛﷯﷮ (𝑥 − 𝑎)﷮2﷯﷯ = 𝑛𝑥﷮𝑛﷯ − 𝑥﷮𝑛﷯ − 𝑛𝑎𝑥﷮𝑛−1﷯+ 𝑎﷮𝑛﷯﷮ (𝑥 − 𝑎)﷮2﷯﷯ = (𝒏 − 𝟏) 𝒙﷮𝒏﷯ − 𝒏𝒂𝒙﷮𝒏−𝟏﷯+ 𝒂﷮𝒏﷯﷮ (𝒙 − 𝒂)﷮𝟐﷯﷯

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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