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 ) py xq= − passes through the point A( )1, 3− where p and q are constants. Given that the gradient of the curve at point A is 9, find the values of p and q. [4] 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 3 of 7 [Ans: p = 648, q = −3] 13 2019/Y4RP/M1/T2/Q4 Given that 2 1(4 1) 3 2 where 1 2yx x x= −− < , find d d y x , simplifying your answer. Hence find the range of values of x for which the gradient of the curve is positive. [5] [Ans: ( )5 4 1 (5 4 ) 32 xx x −− − ; 11 144 x<< ] 14 2019/Y4RP/M2/T1/Q3 Given that e xy= , find the value of k such that 2 2 dd40 dd yyx kyxx + −= . [5] [Ans: k = 2] 15 2019/Y4RP/M2/T1/Q1 Differentiate the following with respect to x, simplifying your answer: (i) 2 sin 3x x , [2] (ii) 23eln cos3 x x   . [2] [Ans: (i) 3 3 cos3 2sin 3xx x x − (ii) 6 3tan 3xx+ ] 16 2019/Y4RP/M2/T1/Q4 (i) By writing sec x as ( ) 1 cos x − , show that ( )d sec sec tand x xxx = . [1] (ii) Hence find the following in terms of sec x , (a) ( )d sec tand xxx ’ [2] (b) ( )d ln sec tand xxx + . [2] [Ans: (ii)(a) ( ) 2sec 2sec 1xx − (b) sec x ] 17 2018/Y4RP/CT/Q1(a) (M1/M2) Differentiate each of the following with respect to x , leaving your answer in the simplest form. (a) ( ) 1234 e xx −+ [2] [Ans: (a) ( ) 1221 4 e xx −−+ ] 18 2018/Y4RP/CT/Q1(b) (M2) Differentiate each of the following with respect to x , leaving your answer in the simplest form. (b) . [2] [Ans: (b) ( ) ( ) 4250 tan 2 sec 2xx ] 19 2018/Y4RP/CT/Q6 (M2) It is given that 1 sinyx= − . (i) Find an expression for d d y x , leaving your answer in the simplest form. . [2] 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 4 of 7 (ii) Show that 2 2 d d y kyx = , where k is a constant to be determined. [4] [Ans: (i) cos 2 1 sin x x − − (ii) 1 1 sin4 x−− ] 20 2017/Y4RP/CT/Q1 (M2) A curve has the equation . Find the gradient of the curve at the point where 1x= . [3] [Ans: −15.1] 21 2017/Y4RP/CT/Q2 (M1/M2) Given that , find , simplifying your answer. [3] [Ans: 2 3 8 (4 1) xxe x+ ] 22 2017/Y4RP/CT/Q3 (M2) Given that , find and , simplifying your answers. Hence show that , where k is a constant to be determined. [5] [Ans: 2 2 dd 2sin4 , 8cos4dd yy xxxx=−= − ] 23 2016/Y4RP/CT/Q1(i) (M1/M2) Differentiate each of the following with respect to x , expressing your answers as a single fraction. (i) 341ln 12 x x + − , [4] [Ans: (i) ( )( ) 23 3 16 8 4 112 xx xx +− +− ] 24 2016/Y4RP/CT/Q1(ii) (M2) Differentiate each of the following with respect to x , expressing your answers as a single fraction. (ii) ( ) 4 2 31 tan 5 x x + . [3] [Ans: (ii) ( ) ( ) 3 2 3 2 3 1 6 tan 5 5 3 1 sec 5 tan 5 x xx x x + −+ ] 25 2016/Y4RP/CT/Q4 (M2) It is given that . (i) Find , leaving your answer in its simplest form. [1] 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 5 of 7 (ii) Show that . [4] [Ans: (i) sin 2 1 cos x x − + ] 26 2015/Y4RP/CT/Q2 (M2) Given that ( )ln 1 sinyx= + , (a) write down an expression for in terms of x, [1] (b) show that . [3] [Ans: (a) d cos d 1 sin yx xx= + ] 27 2015/Y4RP/CT/Q4(a) (M2) For each of the following functions , find , expressing your answers as a single fraction (a) , [4] [Ans: (a) ( ) 21 4 2e cos 4 6sin 4 cos 4 x xx x + + ] 28 2015/Y4RP/CT/Q4(b) (M1/M2) For each of the following functions , find , expressing your answers as a single fraction (b) 25 10 1ln 75 xy x  −=  − . [4] [Ans: (b) 26 (10 1)(7 5 )xx−− ] 29 2014/Y4RP/CT/Q1(a) (M2) Find for each of the following functions, expressing your answers as a single fraction. (a) , [3] [Ans: (a) 3 sin 2 cos sin xx x x − ] 30 2014/Y4RP/CT/Q1(b) (M1/M2) Find for each of the following functions, expressing your answers as a single fraction. 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-