Miscellaneous

Chapter 13 Class 11 Limits and Derivatives
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 Using product rule, (uv)’ = u’v + v’u Finding u’ & v’ u = px + q u’ = p × 1 + 0 = p v = rx – 1 + s v’ = r( – 1) x–1 – 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 − 𝑝𝑟𝑥/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 + qr (–1 x – 1 –1) + 0 + 0 = ps – qr (x-2) = ps – 𝑞𝑟/𝑥^2 Hence, f’(x) = ps – 𝒒𝒓/𝒙^𝟐

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#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.