RI Differentiation C6A Add Prac (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 _______________________________________________ Additional Practice C6A: Differentiation Techniques Page 1 of 2 C6A: Differentiation Techniques (Additional Practice) 1 Differentiate each of the following with respect to x: (a) 22(1 ) e xx (b) cos 4x x   (c) 1 1s i n x (d) 12ln tan ( ) xx     (e) 2 2 1ln 1 x x   (f) lnexx (g) 1tan 3e x (h) 23sec 2xx (i) cot cos ec 2xx (j) 1sin (tan )x (k) 21cos (e )xx  (l) xsin x 2 Find 21d 99 s i nd3 xxxx     , leaving your answer in the form 2ab x , where the values of a and b are to be found. 3 For each of the following, find d d y x in terms of x and y. (a) 33 230xyx y (b) tan 0xx y (c) cos( ) ln( ) 0xy x y 

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ _______________________________________________ Additional Practice C6A: Differentiation Techniques Page 2 of 2 4 It is given that 21 3tan 4xy y  . Find d d y x in terms of x and y. [4] Hence, find the exact value of d d y x when y = 1, given that x > 0. [3] 5 (a) Find d d y x if (i) 3 21y x  (ii) 2 1ln 1c o s xxy x     (b) Given that 3e(5 )yyx y , find d d y x in terms of x and y. 6 A curve C has equation 22 22 41 100 2 xy xx y   , x  , 8.x Show that 2d2 d2 1 6 yx y x xy y   . [2] 7 If sin 2sinyx , show that . 8 (a) Given that 1tan ,yx  find d d y x . [2] (b) Given that ඥ𝑦ೣ ൌ √𝑥 ೤ , where 𝑥൐ 0, 𝑦൐ 0, find d d y x . [4] 9 Given the curve with equation 22 431 0xx y y  , find the gradient of the curve at the point where the y coordinate is 1. 10 (a) Find an expression for d d y x in terms of t for 2 2 22 88, 2 txt y tt  , where 0t  . (b) A curve C has parametric equations 2 2 πs i n2 , c o s 1 , 0 .xt yt t    Express d d y x in the form tan ,ab t where a and b are constants to be determined. [2] (c) The parametric equations of a curve C are 212sin (1 ), e ttxt y   . Find d d y x in terms of t . [4] d d 1+3 s e c 2 2y x y    