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
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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 = 2 Now , f (x) = (uv) = u v + v u = p(rx 1 + s) + 2 (px + q) = p + 2 (px + q) = + ps 2 2 = + ps 2 = 0 + ps 2 = ps 2 Hence, f (x) = ps 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 2 Hence f (x) = ps 2

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