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Misc 3 - Find derivative: (px + q) (r/x + s) - Class 11 CBSE - 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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Misc 3 (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): (px + q) r﷮x﷯+s﷯ Let f(x) = (px + q) r﷮x﷯+s﷯ = (px + q) (rx –1 + s) Let u = (px + q) & v = (rx-1 + s) ∴ f(x) = uv So, f’(x) = (uv)’ f’(x) = u’v + v’u Finding u’ & v’ u = px + q u’ = p 1 . x1-1 + 0 = px0 = p v = rx – 1 + s v’ = r( – 1) x1 – 1 + 0 = – rx – 2 = − 𝑟﷮ 𝑥﷮2﷯﷯ Now , f’(x) = (uv)’ = u’v + v’u = p(rx – 1 + s) + − 𝑟﷮ 𝑥﷮2﷯﷯﷯ (px + q) = p 𝑟﷮𝑥﷯+𝑠﷯ − 𝑟﷮ 𝑥﷮2﷯﷯ (px + q) = 𝑝𝑟﷮𝑥﷯ + ps − 𝑝𝑟 𝑥﷮ 𝑥﷮2﷯﷯ – 𝑟𝑞﷮ 𝑥﷮2﷯﷯ = 𝑝𝑟﷮𝑥﷯ – 𝑝𝑟﷮𝑥﷯ + ps – 𝑟𝑞﷮ 𝑥﷮2﷯﷯ = 0 + ps – 𝑟𝑞﷮ 𝑥﷮2﷯﷯ = ps – 𝑟𝑞﷮ 𝑥﷮2﷯﷯ Hence, f’(x) = ps – 𝒒𝒓﷮ 𝒙﷮𝟐﷯﷯ Misc 3 (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): (px + q) r﷮x﷯+s﷯ Let f(x) = (px + q) r﷮x﷯+s﷯ = px r﷮x﷯+s﷯ + q r﷮x﷯+s﷯ = px r﷮x﷯﷯ + px (s) + q r﷮x﷯﷯ + qs = pr + pxs + qr﷮x﷯ + qs = pxs + qrx-1 + qs + ps Now, So, f’(x) = pxs + qrx−1 + qs + ps﷯﷮′﷯ = (pxs)’ + (qrx-1)’ + (qs)’ + (ps)’ = ps . (1 . x1–1 ) + qr (-1 . x – 1 –1) + 0 + 0 = ps (x0) + qr ( -1 x-2) = ps (1) – qr x –2 = ps – 𝑞 𝑟﷮ 𝑥﷮2﷯﷯ Hence f’(x) = ps – 𝑞𝑟﷮ 𝑥﷮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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