Raffles Institution 2023 Y4 SUPP WS Differentiation

Page 1 of 7 RAFFLES INSTITUTION MATHEMATICS DEPARTMENT 2023 YEAR 4 RP MATHEMATICS TOPIC 3: DIFFERENTIATION (MATHS 1 & MATHS 2) SUPPLEMENTARY WORKSHEET Name: Class: Sec 4 ( ) Date: 1 2021/Y4RP/M1/T2/Q2 (a) Find the derivative of ( )( ) ( ) 633 ttt tee e − + . [2] (b) Given that ( ) 4 5 1ln (2 3) xy x −= − , find d d y x , leaving your answer as a simplified single fraction. [3] [Ans: (a) 273 376 tte te− + (b) 37 ( 1)(2 3) x xx − −− ] 2 2021/Y4RP/M1/T2/Q3 Given that ( ) 22 31xye x= − , prove that 2 2 2 dd 4 46dd xyy yexx − += . [5] 3 2021/Y4RP/M1/T2/Q4 Find the x-coordinate(s) of the point(s) on the curve, 5 3 12 2 yx x x = +   where x > 0, at which the tangent at the point is perpendicular to the line 21xy π−= − , leaving your answer correct to 3 significant figures. [5] [Ans: x=0.408] 4 2021/Y4RP/M1/T2/Q5 The equation of a curve is given by ( ) 2 53 1where 212 xyx x += < − . (i) Find d d y x , giving your answer as a simplified single fraction. [3] (ii) Find the range of values of x for which d d y x is non-negative. [3] [Ans: (i) ( )( ) 3 5 3 13 15 (1 2 ) xx x +− − (ii) 31 52 x−≤< ] 5 2021/Y4RP/M2/T1/Q1 A curve has the equation ( ) 31 tan 2 7 ln sin3yxx= − . Obtain an expression of d d y x in terms of tangent function. [3] [Ans: 42 72 tan 2 2 tan 2 tanxx x+− ] This study source was downloaded by 100000857398866 from CourseHero.com on 11-19-2024 22:50:04 GMT -06:00 https://www.coursehero.com/file/242024773/03-2023-Y4-SUPP-WS-Differentiation-230512-100342pdf/

Page 2 of 7 6 2021/Y4RP/M2/T1/Q2 Given that 2 cos 2 x xy e= , show that 2 2 dd 4dd yy kyxx += , where k is a constant to be determined. [5] [Ans: 8k =− ] 7 2020/Y4RP/M1/T1/Q5 Differentiate the following expressions with respect to x, simplifying your answer. (a) 249x x + [2] (b) ( ) 3 23 41 x x + + [3] (c) ( ) ( )32 5 13xx−− [3] [Ans: (a) 2 3 3(4 3) 2 x x − (b) ( ) ( ) 4 2 8 17 41 x x +− + (c) 23 78 (1 3 ) x x − − ] 8 2020/Y4RP/M2/T1/Q2 Differentiate ( ) 25e ln 2 x x with respect to x, leaving your answer in the simplest form. [3] [Ans: ( )( ) ( ) 252 2 e 10 ln 2 1 ln 2 x xx xx −  ] 9 2020/Y4RP/M2/T1/Q1 Differentiate ( ) 5ln cos 3 x with respect to x, leaving your answer as a single trigonometric term. [2] [Ans: 15 tan 3x− ] 10 2020/Y4RP/M2/T1/Q3 Given that tan 5yx= , show that 2 2 d d y x can be written in the form d d yky x , where k is a constant to be determined. [4] [Ans: k = 10] 11 2019/Y4RP/M1/T2/Q1 (a) Differentiate 72 2 xx − with respect to x. [2] (b) Differentiate 32 56 x x + − with respect to x, simplifying your answer as a single fraction. [3] [Ans: (a) 5 17 4 x x − (b) 3 15 46 2 (5 6) x x − − ] 12 2019/Y4RP/M1/T2/Q3 The curve 23(3