RI Differentiation C6A Tut (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ______________________________________ Tutorial C6A: Differentiation Techniques Page 1 of 3 Tutorial C6A: Differentiation Techniques Section A (Basic Questions) 1 Differentiate each of the following with respect to x: (a) 1 3 3(1 )xx  (b) 3 (3 ) (1 ) x xx (c) 3sec (2 3)x (d) 1sin 2 x   (e) 13cos x giving your answer in non-trigonometric form, (f) 1 1tan 1 x x     (g)   222eexx  (h)  33 22ln ( 2) ( 2)xx (i) 4 2log (3 e ) xx  (j) 2 7x 2 Find an expression for d d y x in terms of x and y for each of the following. (a) sin 2xy y  (b)   2sin ( )yx y (c) 1sin cos 2xy  3 Find an expression for d d y x in terms of t for each of the following. (a) 23 32, 11 ttxy tt   (b) 11(e e ), (e +e )22 tt t txy  

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ __________________________________ Tutorial C6A: Differentiation Techniques Page 2 of 3 Section B (Discussion Questions) 1 By considering the derivative as a limit,   0 ff ( )f' ( ) l i m x x xxx x      , show that the derivative of 2 1 x is 3 2 x . [3] 2 Differentiate each of the following with respect to x: (a) 234x x (b) 2 3 74xx x  (c) 2 (2 ) (1 ) xx x   (d) cos x (e) 42sin x (f) 12cos 1 x  (g) 1sec3 sin 2x x (h)  12tan e x (i)   2cosec ln 3x  (j) 2et a nx x (k)   3ln 9 3e x (l) 2ln 1x  (m) 3 eln cos x x   , where 0 2x  (n) 2 2 1ln , 3 x x a   where a is a positive constant (o) sec2 xx (p)  2 1 x x 

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ __________________________________ Tutorial C6A: Differentiation Techniques Page 3 of 3 Implicit Differentiation 3 (a) Given 23lnnx yy n , where n is a constant, find d d y x in terms of n, x and y. [3] (b) Given that 23 2 e eyxy  , where 1ey  , show that 32 2 d2 d d3 d yy x x y   . [3] (c) It is given that 2 sin cosyx x . Show that 32 32 dd dd 0dd dd yy yyyA yxx xx  , where A is a real constant to be determined. [2] (d) Given that 1 ln xxy x , find d d y x in terms of x and y. [3] 4 Find an expression for d d y x in terms of t for each of the following. (a) sec , tanx at y at , where a is a non-zero constant (b) 2 , , for 1.11 uuxy uuu  5 A curve C has equation 22 22 11 .32 xy yx   (a) Show that 2 2 d2 .d( 3 2 ) yx x y x yx   [3] (b) Find the equation of the tangents to the curve that are parallel to the y-axis. [3] 6 A curve C has parametric equations 22 1 ,, 111 txyt tt   . (i) Find d d y x in terms of t. [2] (ii) Show that there are no points on C with tangents inclining at an angle of 4  with the positive x-axis. [3]