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D. Solve the derivative of the following: 16. y = V2x + 1. 17. f(x) = 2x-3 3x+4 C 18. y = ax+b 19. P(x) = 3sec(x)-10cot(x) 20. h(w) = 3w 4-w2tan(w) 21. y = 5sin(x)cos(x) + 4csc(x) 22. P(t) = sin(t) 3-2cos(t) 5ex 23. y = 3e*+1 24. A(x) = In(x 4 + x4) 25. f(x) = sec(1 - 5x) 26. Z(t) = cost(t) + cos(t4) 27. D(Z) = ( (22+3 3 6-2.2\f16) if f ( 7 ) = x5 4 = ( 2 x + 1) /2 f' ( x ) = n x 7 - 1 4 = 1 ( 2 x + 1 ) . ( 2 ( 1 ) +0 ) 1 - 1 8 = 4 ( 2x + 1 ) 2 ( 2 ) we don't have need to use product rule 4 = 1 and Quotient rule here 12 x + 1 17) f ( x ) = 2x -3 3x + 4 It is of the form f (x ) = U(x ) V (x ) P ( x ) = u ( x ) = *. U' ( x ) v ( xe ) - U ( xx ) v ' (x ) v ( x ) ( V ( x ) ) 2 u ( xx) = 2x - 3 7 u'(x) = 2(1)-0 - 2 V (x ) = 3 x + Lp = v' (x ) = 3 ( 1 ) to = 3 f ( x ) = UI ( x ) V (R ) - UCx ) v' ( x ) ( V (x ) ) ? 2 ( 3x + 4 ) - 3 ( 2x - 3 ) we have used ( 3 * + 4 ) 2 Quotient rule here 8 ( x ) = 6X+8-6# +9 ( 3 x + 4 ) 2 f ( x ) = 17 ( 3 x+ 4 )\f8 ' ( w) = m' (w) ke (w ) +thew ) k' (w) r " ( w ) = 2 w k' (w ) = sectw product rude g (w ) = 2w tanw + w sec"w is used how? = p" (w ) - g"(w ) hew) = _ 12ws - [ 2 w tanw + w sec"w) 21 ) y = 5 sinx cosx + 4 (sc (x) Let p (x ) = 5 sinecosine P(x ) is of the form f ( x). g (x) Product P(x ) = 5 sina 9 ( x ) = cosx rule P ' ( xx ) = ( 8 ( xx ) g ( x ) ) " = p " ( xx ) g ( 2 ) + g' ( xx ) f ( x ) f ' ( x ) = 5 cosx 8 ( 21 ) = - sinx P' ( x ) = SEOSIE (-sinn ) +. 5 cosa cosal + (-sina) 5 sinn P' ( x ) = 5cost - 5 six q ( x ) = - 4 csex cot x ly' = 5 [ cos x - sin x] - 4 (sex cotx 22 7 PCE ) = SME 3 - 2 cost It is of the form P(L) = P(t) get ) Quotient P ( t ) = ( fit ) ) rule get ) = P ( t ) 8 (t ) - 8 ' (t ) f (t ) ( get ) ) 2f (t ) = sint p" ( t ) = cost g ( t ) = 3 - 2 cost g ' ct ) = 0 - 2 ( - sint ) = 2 sint p' (t ) = cost ( 3- 2 cost ) - 2 sint ( sint ) 209 ( 3 - 2 cost ) 2 p' (t ) = 3cost - 2 cos't - 25in' t (3 - 2 cost )? pict ) = 3cost - 2 ( cost + sin> t ) we know cost + sight = 1 ( 3 - 2 cost ) 2 pict ) = 3cost - 2 ( 3 - 2 cost ) 2 23) 4 = sex 3 et1 It is of the form y= f(x) g (x ) y' = f (x ) = Pledgex) - graf(x). ~ Quotient rule 18 (x ) ( 8 ( x ) ) 2 P'( x ) = 5 em 8 ( x ) = 3e" + 1 g' ( x ) = 3ex to y'= sex ( 3e" +1) - 30* (50* ) ( 3 e * + 1) 2 y = 19/ex + 5 ex - 15 6 2x (3e * + 1)2 y = 5 ex (3e " + 1 )24) A(x ) = In ( 34 + 24 ) -4 - 1 4 A ' ( x ) = 1 ( - 4 x + 4 2c - 4 ( x + 24) A ( x ) 1 [-42 + 4x3] ( X 4 x 4 ) no rule is used ( not requay ) A " (x ) = 4 [ 23 - 5 ) ( x 7 + x 4 ) 25) f (x ) = sec (1-5 x ) * if Ph ) = secx' then flex) = secxtank and davivalue of x is, so flex) = secx tann here f (x ) = seco ( 1-5 x ) flew = sec (1- 5 x ) tan (1- 5x ) " (1-5x) ( 1 - 5X ) =0- 5( 1) 2 - 5 P (x ) = sec (1 - 5x) tan (1- 5x) (-5) 8 ( x ) = - 5 sec ( 1- 5x ) tan ( 1-5x ) 26) 2(t ) = cost (t ) + cos( ty) It is to of the form z (t ) = f(t ) + gct) fit ) = cos (E ) f ' ( + ) = 4 cost ) . [ cost ) ! = 4 cost ( - sint ) = - 4 sint cost9 ( 4 ) - cos ( 1 4 ) g'(t ) = - sin (ty ). ( + 4 )' 8 ( ) - - sin ( + 4 ) . 4+ 4 - 1 no jule S necessaly 8' ct ) = - 4 sin( 1 4 ) . + 3 2 " (+ ) = Plet ) + gict ) 2 ' (t ) = - 4 sint cost - 4 sin (14). 13 D ( 2 ) = 2 2+3 3 6 - 22 It is of the form D (Z ) = f ( z ) 8 (2 ) 3 - 1 D ' ( 2 ) = 3 ( f( 2 ) ) . ( f ( 2 ) 8 ( z ) f (z ) = 27+3 9 ( 2 ) = 6- 22" f ( z ) = 2 ( 1 ) to 8 ( 2 ) = 0 - 22 = 2 = -2z - Quotient f ( z ) p ' ( z ) g ( z ) - 8' ( z ) f (z ) rule g (z ) (8 ( z) ) 2 2 ( 6 - 2 2 ) - (- 22) ( 22+3 ) ( 6- 2 2) 2 = 12 - 22 2+ 42 2+62 12 + 2 2 + 62 ( 6- 22 ) 2 ( 6- 22)2 P ( 2 ) = 3 2 22+3 12 + 2 2 +62 6-72 ( 6 - 2 2)2