Дифференциальное исчисление. - 8 стр.

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8 §1. ðÒÏÉÚ×ÏÄÎÁÑ ÆÕÎËÃÉÉ
òÅÛÅÎÉÅ.
arctg
2
e
−x
0
=
arctg e
−x
2
0
= 2 arctg e
−x
·
arctg e
−x
0
=
= 2 arctg e
−x
·
1
1 + (e
−x
)
2
·
e
−x
0
= 2 arctg e
−x
·
1
1 + e
−2x
· e
−x
· (−x)
0
=
= 2 arctg e
−x
·
1
1 + e
−2x
· e
−x
· (−1) = −
2e
−x
arctg e
−x
1 + e
−2x
.
ðÒÉÍÅÒ 20. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ 2 ln
3
4
√
tg 5x.
òÅÛÅÎÉÅ.
2 ln
3
4
p
tg 5x
0
= 2
ln
3
4
p
tg 5x
0
= 2
ln
4
p
tg 5x
3
0
=
= 2 ·
3
ln
4
p
tg 5x
2
·
ln
4
p
tg 5x
0
= 6 ln
2
4
p
tg 5x ·
ln
4
p
tg 5x
0
=
= 6 ln
2
4
p
tg 5x ·
1
4
√
tg 5x
·
4
p
tg 5x
0
=
6 ln
2
4
√
tg 5x
4
√
tg 5x
·
(tg 5x)
1
4
0
=
=
6 ln
2
4
√
tg 5x
4
√
tg 5x
·
1
4
· (tg 5x)
−
3
4
· (tg 5x)
0
=
6 ln
2
4
√
tg 5x
4(tg 5x)
1
4
(tg 5x)
3
4
·
1
cos
2
5x
· (5x)
0
=
=
3 ln
2
4
√
tg 5x
4 tg 5x cos
2
5x
· 5 =
15 ln
2
4
√
tg 5x
4 sin 5x cos 5x
=
15 ln
2
4
√
tg 5x
2 sin 10x
.
úÁÍÅÞÁÎÉÅ. ÷ ÎÅËÏÔÏÒÙÈ ÓÌÕÞÁÑÈ ÕÄÏÂÎÅÅ ÕÐÒÏÓÔÉÔØ ÆÕÎËÃÉÀ ÄÏ ×ÚÑ-
ÔÉÑ ÐÒÏÉÚ×ÏÄÎÏÊ. åÓÌÉ × ÐÒÉÍÅÒÅ 20 ÓÎÁÞÁÌÁ ÐÒÅÏÂÒÁÚÏ×ÁÔØ ÆÕÎËÃÉÀ ÓÌÅ-
ÄÕÀÝÉÍ ÏÂÒÁÚÏÍ, ÔÏ ×ÙÞÉÓÌÅÎÉÑ ÂÕÄÕÔ ÐÒÏÝÅ:
2 ln
3
4
p
tg 5x = 2
ln
4
p
tg 5x
3
= 2
ln (tg 5x)
1
4
3
=
= 2
1
4
ln tg 5x
3
=
1
32
(ln tg 5x)
3
.
ðÒÉÍÅÒ 21. ÷ÙÞÉÓÌÉÔØ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ y(x) =
sin 14x
2 cos 7x
.
òÅÛÅÎÉÅ. óÎÁÞÁÌÁ ÐÒÅÏÂÒÁÚÕÅÍ ÆÕÎËÃÉÀ y(x):
y(x) =
sin 14x
2 cos 7x
=
2 sin 7x cos 7x
2 cos 7x
= sin 7x.
ôÅÐÅÒØ ÎÁÈÏÄÉÍ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ y(x):
y
0
(x) = (sin 7x)
0
= cos 7x ·(7x)
0
= 7 cos 7x.
8                                                                        §1. ðÒÏÉÚ×ÏÄÎÁÑ ÆÕÎËÃÉÉ

     òÅÛÅÎÉÅ.
            2 −x 0
                                          0
                                        −x 2
                                                                                   0
                                                 = 2 arctg e−x · arctg e−x
                
     arctg e          =       arctg e                                                   =
                                 1               −x 0                             1
       = 2 arctg e−x ·                                                 −x
                                                                                          · e−x · (−x)0 =
                                                     
                                       2  ·    e       =   2 arctg   e    ·         −2x
                            1 + (e−x )                                       1+e
                                                  −x         1          −x                  2e−x arctg e−x
                                 = 2 arctg e ·                      · e · (−1) = −                             .
                                                        1 + e−2x                                 1 + e−2x
                                                                               √
     ðÒÉÍÅÒ 20. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ 2 ln3 4 tg 5x.
     òÅÛÅÎÉÅ.
                     0         p               0                         3 0
           3 4                     3 4
             p                                                    p4
      2 ln     tg 5x = 2 ln             tg 5x = 2 ln tg 5x                           =
                                2   p                  0                           p             0
                                                                        2 4
                        p4                         4
                                                                          p                  4
        = 2 · 3 ln tg 5x                · ln tg 5x = 6 ln                     tg 5x · ln tg 5x =

                                    1          p          0 6 ln2 √      4
                                                                             tg 5x                1 0
                                                                                                     
                  2 4
                    p
           = 6 ln      tg 5x · √            ·     4
                                                     tg 5x = √                       · (tg 5x)     4     =
                                 4
                                   tg 5x                               4
                                                                         tg 5x
               √                                                       2√
        6 ln2 4 tg 5x 1                −43
                                                         0        6 ln    4
                                                                             tg 5x            1
    = √                 ·    · (tg 5x)       · (tg   5x)   =              1           3 ·      2
                                                                                                      · (5x)0 =
            4
              tg 5x        4                                   4(tg 5x) 4 (tg 5x) 4 cos 5x
                                                √                             √                        √
                                       3 ln2 4 tg 5x                 15 ln2 4 tg 5x          15 ln2 4 tg 5x
                                  =                          ·5=                          =                    .
                                     4 tg 5x cos2 5x                4 sin 5x cos 5x              2 sin 10x
  úÁÍÅÞÁÎÉÅ. ÷ ÎÅËÏÔÏÒÙÈ ÓÌÕÞÁÑÈ ÕÄÏÂÎÅÅ ÕÐÒÏÓÔÉÔØ ÆÕÎËÃÉÀ ÄÏ ×ÚÑ-
ÔÉÑ ÐÒÏÉÚ×ÏÄÎÏÊ. åÓÌÉ × ÐÒÉÍÅÒÅ 20 ÓÎÁÞÁÌÁ ÐÒÅÏÂÒÁÚÏ×ÁÔØ ÆÕÎËÃÉÀ ÓÌÅ-
ÄÕÀÝÉÍ ÏÂÒÁÚÏÍ, ÔÏ ×ÙÞÉÓÌÅÎÉÑ ÂÕÄÕÔ ÐÒÏÝÅ:
                          p      3             1 3
                                                    
        3
            p
            4               4
    2 ln        tg 5x = 2 ln tg 5x = 2 ln (tg 5x) 4    =
                                                              3
                                                     1            1
                                             =2        ln tg 5x =    (ln tg 5x)3 .
                                                     4            32
                                                                                         sin 14x
     ðÒÉÍÅÒ 21. ÷ÙÞÉÓÌÉÔØ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ y(x) =                                     2 cos 7x .
     òÅÛÅÎÉÅ. óÎÁÞÁÌÁ ÐÒÅÏÂÒÁÚÕÅÍ ÆÕÎËÃÉÀ y(x):
                                      sin 14x    2 sin 7x cos 7x
                           y(x) =              =                 = sin 7x.
                                      2 cos 7x       2 cos 7x
ôÅÐÅÒØ ÎÁÈÏÄÉÍ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ y(x):
                           y 0 (x) = (sin 7x)0 = cos 7x · (7x)0 = 7 cos 7x.

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