Дифференциальное исчисление. - 6 стр.

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6 §1. ðÒÏÉÚ×ÏÄÎÁÑ ÆÕÎËÃÉÉ
ðÒÉÍÅÒ 11. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ f (x) =
x
13
arcctg x
lg x
.
òÅÛÅÎÉÅ. æÕÎËÃÉÑ lg x ¡ ÜÔÏ ÄÅÓÑÔÉÞÎÙÊ ÌÏÇÁÒÉÆÍ, ÔÏ ÅÓÔØ lg x =
= log
10
x. ðÒÉÍÅÎÉÍ ÆÏÒÍÕÌÙ ÐÒÏÉÚ×ÏÄÎÙÈ ÞÁÓÔÎÏÇÏ É ÐÒÏÉÚ×ÅÄÅÎÉÑ:
x
13
arcctg x
lg x
0
=
x
13
arcctg x
0
· lg x −
x
13
arcctg x
· (lg x)
0
(lg x)
2
=
=
x
13
0
arcctg x + x
13
(arcctg x)
0
· lg x −
x
13
arcctg x
·
1
x ln 10
lg
2
x
=
=
13x
12
arcctg x + x
13
−
1
1+x
2

· lg x − x
13
arcctg x
1
x ln 10
lg
2
x
=
=
13x
12
arcctg x lg x −
x
13
lg x
1+x
2
−
x
12
arcctg x
ln 10
lg
2
x
.
1.4. ðÒÏÉÚ×ÏÄÎÁÑ ÓÌÏÖÎÏÊ ÆÕÎËÃÉÉ
ðÕÓÔØ ÆÕÎËÃÉÑ u = ϕ(x) ÉÍÅÅÔ ÐÒÏÉÚ×ÏÄÎÕÀ × ÎÅËÏÔÏÒÏÊ ÔÏÞËÅ x = x
0
,
Á ÆÕÎËÃÉÑ y = f(u) ÉÍÅÅÔ ÐÒÏÉÚ×ÏÄÎÕÀ × ÔÏÞËÅ u
0
= ϕ(x
0
). ôÏÇÄÁ, ÓÌÏÖÎÁÑ
ÆÕÎËÃÉÑ f (ϕ(x)) ÉÍÅÅÔ ÐÒÏÉÚ×ÏÄÎÕÀ × ÔÏÞËÅ x = x
0
, ËÏÔÏÒÁÑ ×ÙÞÉÓÌÑÅÔÓÑ
ÐÏ ÆÏÒÍÕÌÅ
[f(ϕ(x
0
))]
0
= f
0
(u
0
) · ϕ
0
(x
0
).
äÌÑ ËÒÁÔËÏÓÔÉ ÉÓÐÏÌØÚÕÅÔÓÑ ÓÌÅÄÕÀÝÁÑ ÚÁÐÉÓØ ÐÏÓÌÅÄÎÅÊ ÆÏÒÍÕÌÙ:
y
0
x
= y
0
u
· u
0
x
.
ðÒÉÍÅÒ 12. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ ln sin x.
òÅÛÅÎÉÅ. ïÂÏÚÎÁÞÉÍ y = ln u, u = sin x, ÔÏÇÄÁ y = ln sin x. ðÏ ÔÅÏÒÅÍÅ
Ï ÐÒÏÉÚ×ÏÄÎÏÊ ÓÌÏÖÎÏÊ ÆÕÎËÃÉÉ y
0
x
= y
0
u
· u
0
x
. îÁÈÏÄÉÍ:
y
0
u
= (ln u)
0
u
=
1
u
, u
0
x
= (sin x)
0
x
= cos x,
ÏÔËÕÄÁ
(ln sin x)
0
= y
0
x
= y
0
u
· u
0
x
=
1
u
· cos x =
1
sin x
· cos x = ctg x.
ðÒÉÍÅÒ 13. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ y(x) = e
x
2
.
òÅÛÅÎÉÅ. ïÂÏÚÎÁÞÉÍ u = x
2
, ÔÏÇÄÁ y(u) = e
u
. ðÏ ÔÅÏÒÅÍÅ Ï ÐÒÏÉÚ×ÏÄÎÏÊ
ÓÌÏÖÎÏÊ ÆÕÎËÃÉÉ y
0
x
= y
0
u
· u
0
x
. îÁÈÏÄÉÍ:
y
0
u
= (e
u
)
0
u
= e
u
, u
0
x
=
x
2
0
x
= 2x,
6                                                             §1. ðÒÏÉÚ×ÏÄÎÁÑ ÆÕÎËÃÉÉ
                                                                     13
   ðÒÉÍÅÒ 11. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ f (x) = x arcctg                    lg x
                                                                                  x
                                                                                    .
   òÅÛÅÎÉÅ. æÕÎËÃÉÑ lg x ¡ ÜÔÏ ÄÅÓÑÔÉÞÎÙÊ ÌÏÇÁÒÉÆÍ, ÔÏ ÅÓÔØ lg x =
= log10 x. ðÒÉÍÅÎÉÍ ÆÏÒÍÕÌÙ ÐÒÏÉÚ×ÏÄÎÙÈ ÞÁÓÔÎÏÇÏ É ÐÒÏÉÚ×ÅÄÅÎÉÑ:
                                       0
                                                                        · (lg x)0
  13             0       13                             13
                                                                      
    x arcctg x           x    arcctg x    · lg x   −    x    arcctg x
                     =                                                             =
        lg x                                    (lg x)2
                                                    
               13 0                                0
                    arcctg x + x (arcctg x) · lg x − x13 arcctg x · x ln1 10
                                 13
                                                                                
             x
       =                                                                                =
                                             lg2 x
                                              1
                13x12 arcctg x + x13 − 1+x             · lg x − x13 arcctg x x ln1 10
                                                   
                                                 2
           =                                                                          =
                                             lg2 x
                                                                          x13 lg x       x12 arcctg x
                                                  13x12 arcctg x lg x −    1+x2      −       ln 10
                                              =                                                         .
                                                                   lg2 x

1.4. ðÒÏÉÚ×ÏÄÎÁÑ ÓÌÏÖÎÏÊ ÆÕÎËÃÉÉ

   ðÕÓÔØ ÆÕÎËÃÉÑ u = ϕ(x) ÉÍÅÅÔ ÐÒÏÉÚ×ÏÄÎÕÀ × ÎÅËÏÔÏÒÏÊ ÔÏÞËÅ x = x 0,
Á ÆÕÎËÃÉÑ y = f (u) ÉÍÅÅÔ ÐÒÏÉÚ×ÏÄÎÕÀ × ÔÏÞËÅ u0 = ϕ(x0). ôÏÇÄÁ, ÓÌÏÖÎÁÑ
ÆÕÎËÃÉÑ f (ϕ(x)) ÉÍÅÅÔ ÐÒÏÉÚ×ÏÄÎÕÀ × ÔÏÞËÅ x = x0, ËÏÔÏÒÁÑ ×ÙÞÉÓÌÑÅÔÓÑ
ÐÏ ÆÏÒÍÕÌÅ
                        [f (ϕ(x0))]0 = f 0 (u0) · ϕ0 (x0).
äÌÑ ËÒÁÔËÏÓÔÉ ÉÓÐÏÌØÚÕÅÔÓÑ ÓÌÅÄÕÀÝÁÑ ÚÁÐÉÓØ ÐÏÓÌÅÄÎÅÊ ÆÏÒÍÕÌÙ:
                                       yx0 = yu0 · u0x .
   ðÒÉÍÅÒ 12. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ ln sin x.
   òÅÛÅÎÉÅ. ïÂÏÚÎÁÞÉÍ y = ln u, u = sin x, ÔÏÇÄÁ y = ln sin x. ðÏ ÔÅÏÒÅÍÅ
Ï ÐÒÏÉÚ×ÏÄÎÏÊ ÓÌÏÖÎÏÊ ÆÕÎËÃÉÉ yx0 = yu0 · u0x . îÁÈÏÄÉÍ:
                                    1
                    yu0 = (ln u)0u = ,            u0x = (sin x)0x = cos x,
                                    u
ÏÔËÕÄÁ
                                              1             1
            (ln sin x)0 = yx0 = yu0 · u0x =     · cos x =       · cos x = ctg x.
                                              u           sin x
                                                                      2
   ðÒÉÍÅÒ 13. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ y(x) = ex .
   òÅÛÅÎÉÅ. ïÂÏÚÎÁÞÉÍ u = x2 , ÔÏÇÄÁ y(u) = eu . ðÏ ÔÅÏÒÅÍÅ Ï ÐÒÏÉÚ×ÏÄÎÏÊ
ÓÌÏÖÎÏÊ ÆÕÎËÃÉÉ yx0 = yu0 · u0x . îÁÈÏÄÉÍ:
                                                 0
                  yu0 = (eu)0u = eu ,    u0x = x2 x = 2x,

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