RI Differentiation C6A Tut Sect A (Soln)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ______________________________________ Tutorial C6A: Differentiation Techniques Page 1 of 4 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, CJC Promo 9758/2021/Q3(a) (f) 1 1tan 1 x x     (g)  222eexx  (h)  33 22ln ( 2) ( 2)xx (i) 4 2log (3 e ) xx  (j) 2 7x Solution: (a)       2 3 2 3 2 3 2 3 11 33 3 2 33 332 33 3 3 d1 (1 ) (1 ) 1 ( 3 )d3 1 13 1( 3 )3 1 16 33 21 1 xx x x x xx xx x x xx x x                 

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ __________________________________ Tutorial C6A: Differentiation Techniques Page 2 of 4 (b) 22 22 22 2 22 2 22 d3 3 ( 3 ) ( 1 ) ( 2 4 ) 3 d( 3 ) (1 ) ( 3 ) (1 ) 3( 4 3) 6 12 (3 ) (1 ) 93 (3 ) (1 ) 3(3 ) (3 ) (1 ) x xx x x xx x x x x xx x xx x xx x xx               (c)  32 3 d sec (2 3) 3sec (2 3) sec(2 3) tan(2 3) 2d 6sec (2 3) tan(2 3) xx x xx xx        (d)  1 2 2 2 2 d1 1sind2 2 1 12 2 4 1 4 x x x x x          (e)      3 2 1 2 3 2 13 1 2 3 dd cos cosdd 13 21 3 21 xxxx x x x x       (f) 1 2 2 2 22 2 2 d1 1 1 ( 1 ) ( 1 )tand1 ( 1 ) 11 1 (1 ) 2 (1 ) (1 ) (1 ) 1 1 xx x xx x x x x xx x x                  (g)  22 2 222 2 44 d (e e ) 2(e e )(2e +2e )d 4(e e ) x xx x x x xx x      

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ __________________________________ Tutorial C6A: Differentiation Techniques Page 3 of 4 (h)   33 22 3 2 2 32 2 32 42 32 dd ln(( 2) ( 2) ) 3ln( 2) 2ln( 2)dd 3(3 ) 2(2 ) 22 94 22 13 18 8 (2 ) (2 ) xx x xxx xx xx xx xx xx x xx           (i) Let 4 2log (3 e ) xyx 4ln(3 e ) ln 2 xx   3 4 d1 2 e dl n 2 ( 3 e ) x x yx xx   Recall: lnlog for , 0ln a bba b a (j) Let 2 7xy 2 2 ln 7 ln 7 e e x x   2 2 ln 7d 2( l n 7 ) ed 2 (ln 7)7 x x y xx x   Use implicit differentiation 2 2 7 ln ln 7 xy yx   Differentiate w.r.t ,x  2 1d 2l n 7d d 2l n 7 7d x y xyx y xx   Recall formula:     22 f( ) f( )d f' ( ) l nd d 72 l n 7 7d x x xx ax a ax xx   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  Solution: (a) sin 2xy y  Differentiate w.r.t x, dd1c o s 0dd d1 d1 c o s yy yxx y xy    

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ __________________________________ Tutorial C6A: Differentiation Techniques Page 4 of 4 (b)   2sin ( )yx y Differentiate w.r.t x,      2 22 2 2 dd cos 2( ) 1dd dd c o s2 ( ) c o s2 ( )dd 2( ) cosd d 12 ( ) c o s yy xy xyxx yy x yx y x yx yxx xy xyy x xy xy                 (c) 1sin cos 2xy  0Dcif sfer dcos o sin sientiate w.r.t d,n yyxx x y x    dc o s c o s cot cotds i n s i n yx y x yxx y 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   Solution: (a) 22 2 2 22 dd 2 3 dd 3 2 d d d() ()dd 1 dd 1 d d d (1 )( 3) (2 3 ) (1 )(2) (3 2 ) 1 5 [] [](1 ) (1 ) (1 ) (1 ) 51 1 (1 ) (1 ) 5 xt yt y y t tt t tt t xt x tt t t tt t t tt               (b) d1 d d1 d dd d(e e ) (e +e ) d2 d d2 d d d d 11 1 1( e + e ) ( ee ) ( ee ) ( e + e )22 2 2 ee e+ e tt t t t t tt tt t t tt tt xy y y t tt tt x t x                