SAJC Check your Understanding Techniques of differentiation teachers

Check your Understanding (Techniques of Differentiation) For Questions1 to 6, differentiate the expressions with respect to x. 1 (a) 325 6 7xx+− (b) 4 2 135xx x−+ (c) 3x x− (d) 42 15 2xx x −+ ( ) 32 2 2 d1a) 5 6 7d 5(3) 6(2) 15 12 xxx xx xx +− =+ =+ ( ) 42 33 3 3 d1b) 3 5d 3(4) 5 ( 2) 212 5 x x xx xx x x − − −+ = − + − = − − 1c) 1 12d 3d xxx − −  = 1 221 32 xx − −+ = 2 13 2 xx + 1d) Let 42 15 2y x x x = − + = 1 42 21 52x x x − −+ 3 3 2d5 4d2 y x x xx − = − − 3 3 54 2 xx x = − − 2 (a) 8(2 1)x− (b) ( ) 3422 353 xx−+ (c) 231x − (e) 2 1 1x + (d) 1 1x− (f) 4 23 x+

( ) 8 7 7 d2a) 2 1d 8(2 1) (2) 16(2 1) xx x x − =− =− ( ) ( ) ( )( ) ( )( ) 342 24 2 3 23 4 2 22 4 2 d22b) 3 5d3 2 (3) 3 5 (4 6 )3 2 4 6 3 5 4 2 3 3 5 xxx x x x x x x x x x x x x  −+ = − + − = − − + = − − + 2c) ( ) 1 2 2d 31d xx − = ( ) ( ) 1 2 21 3 1 62 xx − − = ( ) 1 2 23 3 1xx − − 2d) ( ) 1 2d 1d xx − − = 3 21 ( 1)2 x − −− = 3 2 1 2( 1)x − − 2e) Let ( ) 21 2 1 11yx x −= = ++ ( ) 22d ( 1)( 1) 2d y xxx −= − + = 22 2 ( 1) x x− + 2f) Let ( ) ( ) 1 2 11 442 3 2 3y x x= + = + ( ) 1 2 3 1 4 2d 1 3 23d 4 2 y xxx − −=+   ( ) 1 2 3 1 4 23 238 xx − −=+   3 (a) 22 ( 3)xx − (b) 2(2 1)(5 3)xx−+ (c) (3 4)(6 )xx x −− (d) 241xx x +− 3a) Let ( ) ( ) 2 22 3 2 6 9y x x x x x= − = − + 322 12 18y x x x= − + 2d 6 24 18d y xxx = − + ( ) 2 32 2 d3b) (2 1)(5 3)d d 10 5 6 3d 30 10 6 xxx x x xx xx  −+ = − + − = − +

3c) Let 2(3 4)(6 ) 22 3 24x x x xy xx − − − −== 122 3 24xx −= − − 2d 3 24( 1)d y xx −=− − − 2 24 3x=− 3d) 2d 4 1 d xx x x  +−  3 1 1 2 2 2d 4d x x xx −= + −  = 311 222 116 22x x x −− ++ 4 (a) 11 227e 2e x − − (b) 2 32xxe +− (c) ( ) 2 eexx −−− (d) ( ) 2exx ee + (e) 42 1 3e 5e 2e x x− + (f) 31 e1 e x x+ − 11 22 1 2 1 2 d4a) 7e 2ed 17 e 02 7 e2 x x x x − −  =−  = ( ) ( ) 2 2 32 32 d4b) e d 2 3 e xx xx x x +− +−=+ 4c) ( ) 2d eed xx x −− − = ( ) ( ) 3 2 e e e ex x x x −−−− − + 4d) Let ( ) 2 2 2e =e +ex x x xy e e +=+ 22d 2e +ed xxy x +=

4e) Let 42 1 3e 5e 2e x xy − += 42 11 3e 5e 2e 2e x xx−−=+ 3 1 335ee22 xx+−=+ 3 1 3d 9 5 eed 2 2 xxy x +−=− = ( ) 3 1 31 9e 5e2 xx+−− ( ) 31 2 1 3 1 2 1 3 1 2 1 3 1 d e 14f ) de d eed 2e ( 3)e 2e 3e x x xx xx xx x x + − − − − − − − − − − − −  −  =− =− − − =− + 5 (a) ( )6ln 3 5x− (b) ( ) 2ln 2 3x + (c) 2ln 1x+ (d) 23ln 2 x +  d5a) 6ln(3 5)d 6(3) 35 18 35 xx x x − = − = − ( ) 2 2 2 d5b) ln(2 3)d 1 423 4 23 xx xx x x  + = + = +

  d25c) lnd1 d ln 2 ln( 1)d 10 1 1 1 xx xx x x   + = − + =− + =− + 5d) Let ( ) 1 2223ln ( 2) 3ln 2y x x= + = + ( ) 23 ln 22 x=+ 2 d 3 2 d 2 ( 2) yx xx =  + 2 3 2 x x= + 6 (a) ( ) 32ln 3 5 2xx−+ (b) ( ) 23ln 8 3 7xx +− (c) 3 2 5ln 4x   + (d) 13ln 21 x x − + 6a) Let ( ) 32ln 3 5 2y x x= − + = ( ) 23ln 3 5 2xx−+ 2 d 3(6 5) d 3 5 2 yx x x x −= −+ 6c) Let 3 2 3 2 5ln ln 5 ln 4 4 yx x = = − + + = ( ) 21ln 5 ln 43 x−+ ( ) 2 2 d 1 2 20d 3 4 34 y x x xx x = − =− + + ( ) ( ) ( ) 1 2 23 23 23 2 23 d6b) ln ( 8) (3 7)d d ln( 8) ln 3 7d d1 ln( 8) ln 3 7d2 29 8 2(3 7) xxx xxx xxx xx xx  +− = + + − = + + − =+ +− 6d) Let 13ln 21 xy x −= + ( ) ( )1 ln 1 3 ln 2 12 xx= − − + d 1 3 2 d 2 1 3 2 1 y x x x −=− −+ 1 3 2 2 1 3 2 1xx =− +−+

7 Differentiate with respect to x (i) ln( ) xex − [4] (ii) 22 1 (2 1)x + [2] ( ) 3 2 d 1 1(i) ln 1d 2 2 22 xx x x x e x ex xex ex xe x − = − −  −= − 2 2 2 2 d 1 8(ii) d (2 1) (2 1) x x x x  −= ++ 8 Differentiate each of the following expressions with respect to x. (i) ( ) 4 23 −x , (ii) ( ) x x 2 1− , (iii) ( )x52ln3 − . [3] [3] [3] (i) ( ) ( )( ) ( ) ( ) 4 3 3 Let 3 2 4 3 2 3 12 3 2 yx dy xdx x =− =− =−

(ii) (iii) 9 Differentiate ( ) 2006 ln x with respect to x . [2] ( ) ( ) 2006 20052006ln lnd xxdx x = 10 Differentiate the following with respect to x: (i) 1 21x− , [2] (ii) 331 ()2 x x xe e e−− − , [3] (iii) 2(1 ) 1 x x + − . [3] (i) ( ) 2 12 21 21 d dx x x  =− − − ( ) 2 2 1 2 2 11 )1(1 2 12 1 Let x xdx dy xxy x xxy x xy −= −+= +−= +−= −= − − ( ) ( )x xdx dy xy 52 15 52 53 )52ln(3 Let − −= − −= −=

(ii) (iii) ( ) ( ) ( ) 3 3 4 2 42 42 11 22 1 422 2 x x x x x xx xx dd e e e e edx dx ee ee − − − − −  − = − = − − =− − ( ) 22 2 (1 ) 1 2 11 43 1 4 1 1 d x d x x dx x dx x d xdx x x    + + +=  −−    = − − + − =− −