Дифференциальное исчисление. - 6 стр.

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,