10 Differentiation TutSol

Page 1 of 16 Topic 10: 1 By considering the derivative as a limit, show that the derivative of 3x is 23x . [N00/I/4] [Solution] Let 3f ( )xx . Then         3 2 3 32f ( ) 3 3x x x x x x x x x x                   233 2 333f ( ) f ( ) x x x x x x xx x x xx             22 0 0 0 f ( ) f ( )lim 3 3 lim lim x x x x x x x x x xx           2f ( ) 3xx (shown) Watch out for mistakes with notations. 2 Differentiate each of the following with respect to x simplifying your answer. (a) 2 24 x x (b) 1 x (c) 43 3 1 21 x x     [Ans: (a)     2 3 2 2 8 4 xx x   (b)   1 41 xx  (c)     323 53 36 1 21 xx x   ] [Solution] (a) 2 2 d d 4 x x x            1 2 2 2 2 2 14 2 4 2 2 4 x x x x x x               23 3 2 2 2 3 2 2 24 4 8 4 x x x x xx x       (b) d 1d xx     1 1 1 221 41xx xx   

Page 2 of 16 (c) 43 3 d1 d 2 1 x xx        43 43 1d d 21 x x x                                 4 3 4 33 3 2 3 3 2 83 32 3 3 3 53 323 53 2 1 4 1 3 1 4 2 1 6 21 12 1 2 1 2 1 21 36 1 21 x x x x x x x x x x x x xx x                (Alternative, apply chain rule first) 43 3 d1 d 2 1 x xx    333 33 1 d 14 2 1 d 2 1 xx x x x                     333 33 333 33 33 33 33 13 3 33 232 3 2 1 d 14 2 1 d 2 1 1 1 d 2 24 2 1 2 d 2 1 1 1 d 341 2 1 2 d 2 1 1d2 3 2 12 1 d 12 3 2 1 621 36 xx x x x xx x x x x x x x x xxx x xxx x                                                   33 53 1 21 x x   3 Find the derivative with respect to x of (a) cos ,x (b) 2cot(1 2 ),x (c)   3tan 5 , x (d) sec .1 tan x x [Ans: (a) sin180 x (b)   224 cosec 1 2xx  (c)     2215tan 5 sec 5xx (d)     2 sec tan 1 1 tan xx x   ]

Page 3 of 16 [Solution] (a) Let cos cos 180 xyx   d sin sind 180 180 180 yx xx      (b)       2 2 2 2 2d cot(1 2 ) cosec 1 2 4 4 cosec 1 2d x x x x xx         (c)           3 2 2 2 2d tan 5 3tan 5 sec 5 5 15tan 5 sec 5d x x x x xx     (d) d sec d 1 tan x xx           2 2 1 tan sec tan sec sec 1 tan x x x x x x            22 22 2 2 sec tan tan sec but 1 tan sec 1 tan sec tan 1 1 tan x x x x xx x xx x        4 Find the derivative with respect to x of (a) 1 sin3e xy  (b) 1 2e xyx (c) 2 1ln 1 xy x     (d)  ln 2 xy x (e)   4 2log 3 e xyx (f) ln(sin )3 xy [Ans: (a) 1 sin33e cos3x x (b)   1 e 2 1x x (c)   2 1 11 x xx   (d) 2 1 ln(2 )x x  (e)   3 4 12 e 3 e ln 2