1. Математический анализ (850924), страница 12
Текст из файла (страница 12)
<<=(.1) y = x2:< y = (x2) = 2x2 1 = 2x: >2) y = x5:< y = (x5) = 5x5 1 = 5x4: >0;000000000000000000000000;00;1960000003) y = x:< y = (x) = (x1) = 1 x1 1 = x0 = 1: >000;p4) y = x:p 1=2 1 (1=2) 1 1 1=2< y = x = x = 2x= 2 x = 2p1 x : >5) y = x1 :1! < y = x = x 1 = ;1 x 1 1 = ;x 2 = ; x12 : >3 2xx6) y = 2 + 3 + x ; x62 :01 ! 0 31 ! !3 2xx6< y = @ 2 + 3 + x ; x2 A = x2 + @ x3 A + x2 ; x62 =1!1! 1 1 311= 2 (x) + 3 x + 2 x ; 6 x2 = 2 + 3 3x2 ; 2 x12 ; 2! 1;6 ; x3 = 2 + x2 ; x22 + 12x3 : >00000;;0;; ;0;0000000007) y = (x2 ; x + 3)(x3 + 2x2 + 4):< y = (x2 ; x + 3) (x3 + 2x2 + 4) = (x2 ; x +3) (x3 +2x2 +4)++(x2 ; x + 3) (x 3 + 2x2 + 4) = (x2) ; (x) + (3) (x3 + 2x2 + 4)++(x2 ; x + 3) (x3) + 2(x2) + (4) = (2x ; 1) (x3 + 2x2 + 4)++(x2 ; x + 3) (3x2 + 4x) = 5x4 + 4x3 + 3x2 + 20x ; 4: >0000000000C ( ($ < + )+ $ ! $+ , 1!.
6 < 1 ''*(, , ) $ + , 1!:(uvw) = (uv) w + (uv)w = (u v + uv )w + uvw =)000000(uvw) = u vw + uv w + uvw :1970000? $( $ <%=!, 7.2(3,4,5):1!p 1x = 1 x = 2px x = ; x12 : 7.3. <<=(000y = (x2 + 1)(x2 + 2)(x2 + 3):< y = (x2 + 1)(x2 + 2)(x2 + 3) = (x2 + 1) (x2 + 2)(x2 + 3)++(x2 + 1) (x2 + 2) (x2 + 3) + (x2 + 1)(x2 + 2)(x2 + 3) == 2x (x2 + 2)(x2 + 3) + (x2 + 1) 2x (x2 + 3) + (x2 + 1)(x2 + 2) 2x == 2x((x2 + 2)(x2 + 3) + (x2 + 1) (x2 + 3) + (x2 + 1)(x2 + 2)) == 2x(3x4 + 12x2 + 11): > 7.4. <<=(.1) y = x2 x; 1 : x ! x (x2 ; 1) ; x(x2 ; 1)< y = x2 ; 1 ==(x2 ; 1)22 1 ; x 2xx2 + 1 : >=x ;=;(x2 ; 1)2(x2 ; 1)2000000000cos x :2) y = x +sin x cos x !x) ; cos x(x + sin x) =< y = x + sin x = (cos x) (x + sin(x + sin x)2x) ; cos x(1 + cos x) = ; x sin x + cos x + 1 : >= ; sin x(x + sin(x + sin x)2(x + sin x)200002x:3) y = 1x ;costg x0 212 cos x) (1 ; tg x) ; x2 cos x(1 ; tg x)xcosx(x@A= 1 ; tg x ==(1 ; tg x)2 2222 000<y((x ) cos x + x (cos x) )(1 ; tg x) ; x cos x ;(1= cos x)==(1 ; tg x)21980002 sin x)(1 ; tg x) + (x2 = cos x)(2xcosx;x=:>(1 ; tg x)2arctg x :4) y = arcctgx arctg x !x ; arctg x (arcctg x) =< y = arcctg x = (arctg x) arcctg(arcctgx)2 (1=(1 + x2)) arcctg x ; arctg x ;(1=(1 + x2))==(arcctg x)2x + arctg x == (1arcctg222(1 + x2)(arcctg x)2 + x )(arcctg x)!$ ( ($ 1 arcctg x + arctg x = 2 : >00005) y = x cos x arcsin x:< y = (x cos x arcsin x) = cos x arcsin x ; x sin x arcsin x++x cos x p 1 2 : >1;x6) y = 2x2 lg x:22x2< y = (2x lg x) = 4x lg x + x ln 10 = 4x lg x + ln2x10 : >7) y = (x + 2) log2 x:< y = ((x + 2) log2 x) = log2 x + xx+ln22 : >ln x :8) y = 44 +; ln x 4 + ln x ! (1=x) (4 ; ln x) ; (4 + ln x) (;1=x)=< y = 4 ; ln x =(4 ; ln x)2= x(4 ;8ln x)2 : >9) y = (x2 ; 1)ex:< y = ((x2 ; 1)ex) = 2xex + (x2 ; 1)ex = ex(x2 + 2x ; 1): >10) y = 3x:< y = (3x) = 3x ln 3: >199000000000000(' 2" ! "$ &'"&" :1) y = (x ; x + 7)(x + x ; 2)34p7) y = xln;x2 8) y = x + 3 2 log 2x + 19) y = ex tg x10) y = 6x11) y = x2x :2pp22) y = (1 + x)(1 + 2 x)(1 + 3 x)3) y = xx ++41 4) y = tg x + arccos x5) y = 1 x++sin1 x x6) y = 2 arcsinx221) y = 7x + 28x ; 9x + 14x + 23) y = ;x(x ;+8x1)+ 1 (x + 1) cos x 5) y = 1 + sin(1x +; sinx)) ; ln x 7) y = 1 ; ((2x=x ; 2) 1 9) y = ex tg x + cos x 11) y = x (3 ;2xx ln 2) :603220220202023p2) y = 3 + 11pxx + 9x 4) y = cos1 x ; p 1 p 1;x2 x ; 1 ; x arcsin xp6) y =x 1;x; (x + 3) (2=(x ln 2)) 8) y = 2x(2 log x +(21)logx + 1)10) y = 6x ln 60022202222022020||||{+ 1 " , )+, + + <%= % ') ++ %(% 1), ++ - .( <%=+ = f (x) + %% 1+ <%=+ = (u()) ..
)++ <%= u = u() (1)! ) # <%= = (u): , <%= u(x) y(u) $ ( ux yu ), <%=+ y = f (x) = y(u(x)) $ yx = yuux20000000 , '$)+,dy = dy du :dx du dx + <<=+ 1!<%= + %+ $ , <%=!, , 1. 7.5. <<=( = sin x3:< 8) <%= # = sin u u = x3 .. 1)! x3: @<<= # <%=yu = cos u = cos x30$ {ux = 3x2, 1+ $(, ) $ ! <%=y = yx = cos x3 3x2: >000 7.6. <<=(p = cos :< 8)+ <%= = cos u u = p: @<<=+ %pyu = ; sin u = ; sin x ux = 2p1 x )py = yx = ; sin x 2p1 x : >0000201 7.7.
<<=( = ctg x1 :< C ( <<=+ % 1+ $( $ x1 )!11y = ; sin2(1=x) ; x2 = x2 sin12(1=x) : > 7.8. <<=(0 = sin x:< 6 <%=+ )++ %$( = u u = sin xy = esin x cos x: >0" , 7.9{7.17 <<= + ' . 7.9.
<<=(y = x:p< y = e x 2p1 x : >p0 7.10. <<=( = arcsin ex:< y = p 1 2x ex: >1;e 7.11. <<=( = arctg ln x:< y = ; 1 + 1ln2 x x1 : >20200 7.12. <<=(py = arcctg x:!11< y = 2parcctg x ; 1 + x2 : > 7.13. <<=(0 = ln ch x:< y = ch1x sh x = th x: > 7.14. <<=(0p = sh x:3< y = p 1 2 ch x: >3 sh x 7.15. <<=(03 = tg x2:< y = cos12 x2 2x: > 7.16. <<=(0 = tg2 x:< C ( #++ <%=+ = u2 { +, ++u = tg x { . sin x : >y = 2 tg x cos12 x = 2 cos3x0 7.17.
<<=( = sin4 x:< y = 4 sin3 x cos x: >0203(' 2" ! "$ &'"&" :1) y = ln sin x4) y = cos x2) y = sh(ln x)5) y = ln x3) y = (arccos x) 6) y = sh x:33234) y = ;3 cos x sin x5) y = 3 lnx x 6) y = sh 2x:1) y = ctg xx) 2) y = ch(lnx02003) y = ;3(arccos x) p 1 1;x020202||||{ 7.18. <<=( = sin4 x2:< " *, , ) <%=! $ % <%=+ (#! !), $ %, +++ '). C ( ) <%= $, , y = u4 u = sin x2 )y = 4 sin3 x2 (sin x2) :" #! '= $ ! (sin x2) . <%=u = sin 2 ) ( ' ( %% 1 ) $(+ u = sin v v = x2 )(sin x2) = cos x2 2x:0000$ + $ ! <%=y = 4 sin3 x2 cos x2 2x:0C, ) , + <%=+ $ ( ' ) <%)% $y = u4 u = sin v v = x2204 $ + $ $ , , ,$(y = yx = yu uv vx: >00000 <<=+.
7.19. <<=(p = cos5 x:< @+ + % ') $ <%=+ )++ %1 $:py = u5 u = cos v v = x $ +ppy = 5 cos4 x (; sin x) 2p1 x : >08) <%= $(+ 1 ( %% , !!, % ++ )$)+ <%= $ . %, p +p )+ ) 1! %( $ %p cos x , %=, ( (cos x)5: @<<=')p4$ + ' + %: 5 cos x { $ + ,(; sin px) { $ + %, 2p1 x { $ + %+. 7.20. <<ee=(p = ctg ln x:< ) , + ( <, %(, % . @<<= ' + %:1p1 1 : >y = ; 2psin ln x 2 ln x x0205 7.21. <<ee=(=epsin x5 :< @+ + % ') $ '+ ) ! <%= ) $:py = eu u = v v = sin w w = x5.. ) $ + ' $ $ , , ), $(:y = e sin x p 1 5 cos x5 5x4: >2 sin x ''*+ )! )+ <<=! <%= ' ) $(.p05(' 2" ! "$ &'"&" :r x1) y = sin 5x2) y = cos 2 3) y = log 8log (log x)]:21) y = 5 sin 10x0242) y = ; qsin(x=2) 4 cos(x=2)033) y = log (log x) log1 x x ln 2 ln 3 ln 4 :0344||||{ % $ , ' 1, )+, <<=+ 1! <%= %'+ <<=+.
7.22. <<ee=( = sin(x5 + x3 + 1):< <<=+ 1! <%=y = cos(x5 + x3 + 1) (x5 + x3 + 1) :20600++ <<=+ +!, )y = cos(x5 + x3 + 1) (5x4 + 3x2): >0 7.23. <<ee=(2 = cos x4x+ 1 :< D'+ $ ( <<=+ 1! <%=, ', +!, )4 + 1) ; x2 4x34 ; 1)222x(x2x(xxx= (x4 + 1)2 sin x4 + 1 : >y = ; sin x4 + 1 (x4 + 1)20I ) %'+ <<=+ ,7.24{7.28. 7.24. <<ee=(y = (x3 + 1)7:< y = 7(x3 + 1)6 3x2: > 7.25. <<e=(y = p 21 :x +4< y = ; q 21 3 2x: >2 (x + 4) 7.26.
<<=(y = sin(3x + 7):< y = cos(3x + 7) 3: > 7.27. <<=(p 42y = arctg(x ; 1 + x ):207000< <<=+ 1! <%= 1 <<=+ $(, ) $ + %+ , + *( <<=+ 1!<%=.!113py =;2x ; p 4x : >1 + (x2 ; 1 + x4)22 1 + x40 7.28. <<=(2xsiny = 1 + tg3 x :< ?) ++ <<=+ ', % $ , )+ $+ { <<=+ 1! <%=.3 x) ; sin2 x 3 tg2 x (1= cos2 x)2sinxcosx(1+tg:>y=(1 + tg3 x)20N + *+ %+ $ ! ' $ '$( 1, $ * <%=. 7.29. <<=(vu2u4 x + x+ 1t = ln:x2 ; x + 1< C ( ='$ ($( ! < <%= % y = 14 (ln(x2 + x + 1) ; ln(x2 ; x + 1)): 2x + 1!12x;1y = 4 x2 + x ; 1 ; x2 ; x + 1 : > 7.30. <<=(0ry = (x + th4 x)2:3208< C ( ='$ ($( ! !.y = (x + th4 x) :23 !211y =3q 1 + 4 th3 x ch2 x : >42(x + th x) 7.31.
<<=(4 + 2x2 ; 1x:y=x2< C ( * <<=( 1y = x2 + 2 ; x12 y = 2x + x23 : >030(' 2" ! "$ &'"&" :p1) y = (1 ; 2x) 5) y = sin 3 1 + x +x 6) y = log (x + 2x + 1)2) y = 11 ;x3) y = cos(2x + 1)7) y = arcsin 2xp; 1 :3x;14) y = tg 3 1025231) y = ;20(1 ; 2x) 90+ x) 2) y = 10(1(1 ; x)3) y = ;2 sin(2x + 1)40604) y = 3 cos ((x1; 1)=3)p5) y = cos 3 1 + x q3 2x3 (1 + x )6) y = (x +32xx ++ 21) ln 2 27) y = q:3 ; (2x ; 1)022 22030202||||{209 8 .
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
