TMJC H2 Chapter 7 Differentiation Discussion Questions_Solutions_2024

Chapter 7 Differentiation TMJC 2024 Page 1 of 19 H2 Mathematics (9758) Chapter 7 Differentiation Discussion Solutions Level 1 1 Differentiate the following with respect to the variable given: (a)  cotx x (b)  3ln( )z z (c)   3 log 1 3a x , where a is a positive constant and 1a (d)   2ln sin  (e) sin3x x (f) 1 1 cos 2 x (g) 3sin x (h) 1cosec 3 2x     (i)   1 3tan 1 x  Solutions: (a)    d cotd x xx         2 2 1 1cot cosec 2 2 1 1cot cosec 22 x x x x x x x x      (b)    3d ln( )dz z z         2 3 2 3 13 ln( ) ln( ) 3 ln( ) ln( ) z z z z z z         ALERT:    3ln( ) 3lnz z  (c) First, we do a change of base for   3 log 1 3a x .         3 3 3 e e log 1 3 log 1 3 ln 1 3 3ln 1 3 log ln ln a x x x x a a a          3d log 1 3d a xx  Always simplify ln expression before differentiating Note that a is a constant, 0, 1a a 

Chapter 7 Differentiation TMJC 2024 Page 2 of 19      ln 1 3d3 d ln 3 3 ln 1 3 9 ln 3 1 x x a a x a x            (d)    2d ln sind  2 1 (2sin cos ) sin cos2 sin 2cot          (e) Method 1: Using Implicit Differentiation sin3x x Let sin3x xy ln sin ln 3y x x       sin Differentiate wrt , 1 d cos sin ln3d d (ln3) cos sind d 3 (ln3) cos sind x x x y x x xy x y y x x xx y x x xx       Method 2: By memorising formula   d sin3d x x x   sin3 ln 3 cos sinx x x x x  Note: The power is a variable in x, so we will need to use implicit differentiation here. ALERT: You cannot do the usual way     d sin sin 13 sin 3dx x x x x x x  Replace y by sin3x x Lecture Notes pg 4:   f ( ) f ( )d f ' lnd x xa a x ax 

Chapter 7 Differentiation TMJC 2024 Page 3 of 19 (f) 1 d 1 d cos 2x x                     11 21 2 21 2 21 2 21 2 d cos 2d 11 cos 2 1 2 2 cos 2 1 2 2 cos 2 1 2 (2 ) 2 cos 2 1 4 xx x x x x x x x x                       (g)     3 2 d sind 3sin cos xx x x (h) 1 2 d 1 cosec 3d 2 1 1 1 1cosec 3 cosec 3 cot 3 (3)2 2 2 2 3 1 1cot 3 cosec 32 2 2 xx x x x x x                                                        (i)        1 3 2 23 2 3 6 d tan 1d 1 3 1 1 3 2 2 xx x x x x x          ALERT: 1 1cos cosx x   Therefore, since 1 1 1cos 2 cos 2 1 cos 2cos 2 x x xx     inverse of cosine function reciprocal of cosine function MF26: Note: Since this question is to differentiate 1cos 2 x , remember to chain rule 2x, i.e. differentiate 2x to get 2.  f x  f ' x

Chapter 7 Differentiation TMJC 2024 Page 4 of 19 2 2004/I/14 modified Find the x-coordinates of all the stationary points on the curve   3 2 xy x a   where 1a and sta