Дифференциальное исчисление. - 9 стр.

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§1. ðÒÏÉÚ×ÏÄÎÁÑ ÆÕÎËÃÉÉ 9
1.5. ðÒÏÉÚ×ÏÄÎÁÑ ÓÔÅÐÅÎÎÏ-ÐÏËÁÚÁÔÅÌØÎÏÊ ÆÕÎËÃÉÉ
äÌÑ ×ÙÞÉÓÌÅÎÉÑ ÐÒÏÉÚ×ÏÄÎÏÊ ÆÕÎËÃÉÉ ×ÉÄÁ u(x)
v(x)
ÐÒÅÄ×ÁÒÉÔÅÌØÎÏ ÎÁÄÏ
ÐÒÅÄÓÔÁ×ÉÔØ ÄÁÎÎÕÀ ÆÕÎËÃÉÀ × ×ÉÄÅ
u(x)
v(x)
= e
ln u(x)
v(x)
= e
v(x) ln u(x)
.
ðÒÉÍÅÒ 22. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ y(x) = x
x
.
òÅÛÅÎÉÅ.
y
0
(x) = (x
x
)
0
=
e
ln x
x
0
=
e
x ln x
0
= e
x ln x
(x ln x)
0
=
= x
x
(x ln x)
0
= x
x
[(x
0
ln x + x(ln x)
0
] = x
x
1 · ln x + x ·
1
x
= x
x
(ln x + 1).
úÁÍÅÞÁÎÉÅ. æÕÎËÃÉÑ x
x
ÉÚ ÐÒÉÍÅÒÁ 22 ÎÅ Ñ×ÌÑÅÔÓÑ ÎÉ ÆÕÎËÃÉÅÊ ×ÉÄÁ
x
α
, ÎÉ ÆÕÎËÃÉÅÊ ×ÉÄÁ a
x
, ÐÏÜÔÏÍÕ ÂÕÄÅÔ ÏÛÉÂËÏÊ ×ÙÞÉÓÌÑÔØ ÐÒÏÉÚ×ÏÄÎÕÀ
ÄÁÎÎÏÊ ÆÕÎËÃÉÉ ÏÄÎÉÍ ÉÚ ÓÌÅÄÕÀÝÉÈ ÓÐÏÓÏÂÏ×:
(x
x
)
0
= x · x
x−1
, (x
x
)
0
= x
x
ln x.
ðÒÉÍÅÒ 23. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ y(x) = (sin x)
ln x
.
òÅÛÅÎÉÅ.
y
0
(x) =
(sin x)
ln x
0
=
e
ln(sin x)
ln x
0
=
e
ln x ln sin x
0
=
= e
ln x ln sin x
(ln x ln sin x)
0
= (sin x)
ln x
[(ln x)
0
ln sin x + ln x(ln sin x)
0
] =
= (sin x)
ln x
1
x
ln sin x + ln x
1
sin x
(sin x)
0
=
= (sin x)
ln x
ln sin x
x
+
ln x cos x
sin x
.
úÁÄÁÞÉ ÄÌÑ ÓÁÍÏÓÔÏÑÔÅÌØÎÏÇÏ ÒÅÛÅÎÉÑ
îÁÊÔÉ ÐÏ ÏÐÒÅÄÅÌÅÎÉÀ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ:
1. y = x
2
;
2. y = x
3
;
3. y = x
4
;
4. y =
√
x;
5. y =
1
x
2
;
6. y =
1
x
3
;
7. y =
1
√
x
;
8. y = sin 2x;
9. y = cos
x
2
;
§1. ðÒÏÉÚ×ÏÄÎÁÑ ÆÕÎËÃÉÉ                                                                           9

1.5. ðÒÏÉÚ×ÏÄÎÁÑ ÓÔÅÐÅÎÎÏ-ÐÏËÁÚÁÔÅÌØÎÏÊ ÆÕÎËÃÉÉ

  äÌÑ ×ÙÞÉÓÌÅÎÉÑ ÐÒÏÉÚ×ÏÄÎÏÊ ÆÕÎËÃÉÉ ×ÉÄÁ u(x)v(x) ÐÒÅÄ×ÁÒÉÔÅÌØÎÏ ÎÁÄÏ
ÐÒÅÄÓÔÁ×ÉÔØ ÄÁÎÎÕÀ ÆÕÎËÃÉÀ × ×ÉÄÅ
                                                 v(x)
                           u(x)v(x) = eln u(x)          = ev(x) ln u(x) .
    ðÒÉÍÅÒ 22. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ y(x) = xx .
    òÅÛÅÎÉÅ.
                         x 0      0
 y 0 (x) = (xx )0 = eln x = ex ln x = ex ln x (x ln x)0 =
                                                             
        x         0  x      0         0       x             1
   = x (x ln x) = x [(x ln x + x(ln x) ] = x 1 · ln x + x ·     = xx (ln x + 1).
                                                            x
    úÁÍÅÞÁÎÉÅ. æÕÎËÃÉÑ xx ÉÚ ÐÒÉÍÅÒÁ 22 ÎÅ Ñ×ÌÑÅÔÓÑ ÎÉ ÆÕÎËÃÉÅÊ ×ÉÄÁ
xα , ÎÉ ÆÕÎËÃÉÅÊ ×ÉÄÁ ax , ÐÏÜÔÏÍÕ ÂÕÄÅÔ ÏÛÉÂËÏÊ ×ÙÞÉÓÌÑÔØ ÐÒÏÉÚ×ÏÄÎÕÀ
ÄÁÎÎÏÊ ÆÕÎËÃÉÉ ÏÄÎÉÍ ÉÚ ÓÌÅÄÕÀÝÉÈ ÓÐÏÓÏÂÏ×:
                         (xx )0 = x · xx−1,             (xx )0 = xx ln x.
  ðÒÉÍÅÒ 23. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ y(x) = (sin x)ln x .
  òÅÛÅÎÉÅ.
                                         0
                 ln x 0
  0                         ln(sin x)ln x
                                                             0
                                             = eln x ln sin x =
                     
 y (x) = (sin x)        = e
     = eln x ln sin x (ln x ln sin x)0 = (sin x)ln x [(ln x)0 ln sin x + ln x(ln sin x)0] =
                                                                         
                                   ln x 1                    1          0
                        = (sin x)          ln sin x + ln x       (sin x) =
                                         x                 sin x
                                                                                           
                                                                ln x ln sin x    ln x cos x
                                                     = (sin x)                +               .
                                                                          x         sin x

úÁÄÁÞÉ ÄÌÑ ÓÁÍÏÓÔÏÑÔÅÌØÎÏÇÏ ÒÅÛÅÎÉÑ

îÁÊÔÉ ÐÏ ÏÐÒÅÄÅÌÅÎÉÀ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ:
  1. y = x2;
  2. y = x3;
           4
  3. y = x
         √;
  4. y = x;
  5. y = x12 ;
  6. y = x13 ;
  7. y = √1x ;
  8. y = sin 2x;
  9. y = cos x2 ;

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