Differentiation Ex 2.1 + 2.2, Exercises of Mathematics

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& Punjab Group of Colleges Gujranwala Mathematics Part-II ( EX 2.1 & 2.2) Name: Class/Section: Roll No: Exercise 2.1 Q.1. ‘Find by definition the derivatives w.r.t ‘x’ of the following functions defined as: ; 7 am at ; i (i) 2x?4+1 (ii) 2- a|x (iii) li (iv) xe 1 aw 2 “ ; (v) va vi) x (x — 3) (vii) a (viii) (x+4) 3 > 1 (ix) x2 (x) x2 (xi) x™ (xii) a (xiii) x (xiv) = x7 200 (i) 2x2+1 (LHR 11) Sol: Let f(x) 2 2x7 +1 > f(x + x) = 2(x+ 6x)?4+1 f(x + Sx) - f(x) = 2(x+ dx)? +1-2x°-1 f(x + &x) — f(x) = 2(x + &x)? - 2x? Lim f(x + &x) - f(x) Lim 2(x + 8x)? — 2x? f(x) 6x0 6x ~ 6x30 6x _ Lim 2 [x? + 2x &x + (8x)7] — 2x? 7 6x30 bx Lim 2x? + 4x &x + 2 (6x)? — 2x? = &x>0 Sx _ Lim 4x &x +2 (dx) _ Lim &x(4x +2 3x) ~ bx > 0 5x ~ 8x30 5x _ Lim my _ = 3x > 0 (4X +2 5x) = 4x +0 = 4x (i) 2->/x Sol: Let f(x) = 2-.x = 2-x!”? f(x + 5x) = 2-(x + Sx)? f(x + 5x) - f(x) = 2-(x + 8x)? -2 +x = —(x + Sx)? + x? Lim f(x +8x)- f(x) — Lim -(x + dx)¥? +x"? 8x0 &x ~ 8x30 8x 8x)? 8x)? =x"? (a + *) + xi? ea =x"? (1 2) -1 &x ~ &x>0 dx f'(x) = 1 (6) . ; G . 7 (ey oe i _ yl hal Lim ~~ me 7! \ x eeeets = éx 30 6x 6-4) of 3(89 lee (3% | Lim —x 21x + 21 x eB oes is = $x > 0 8x Scanned with CamScanner