Дифференциальное исчисление. - 7 стр.

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§1. ðÒÏÉÚ×ÏÄÎÁÑ ÆÕÎËÃÉÉ 7
ÏÔËÕÄÁ
y
0
x
=
e
x
2
0
= y
0
u
· u
0
x
= e
u
· 2x = e
x
2
· 2x = 2xe
x
2
.
þÁÓÔÏ ÂÏÌÅÅ ÕÄÏÂÎÏ ÎÅÐÏÓÒÅÄÓÔ×ÅÎÎÏ ÎÁÈÏÄÉÔØ ÐÒÏÉÚ×ÏÄÎÙÅ ÐÒÏÍÅÖÕÔÏÞ-
ÎÙÈ ÆÕÎËÃÉÊ.
ðÒÉÍÅÒ 14. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ ln sin x (ÓÍ. ÐÒÉÍÅÒ 12).
òÅÛÅÎÉÅ.
(ln sin x)
0
=
1
sin x
(sin x)
0
=
1
sin x
cos x = ctg x.
ðÒÉÍÅÒ 15. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ y(x) = e
x
2
(ÓÍ. ÐÒÉÍÅÒ 13).
òÅÛÅÎÉÅ.
y
0
(x) =
e
x
2
0
= e
x
2
· (x
2
)
0
= e
x
2
· 2x = 2xe
x
2
.
ðÒÉÍÅÒ 16. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ e
−x
.
òÅÛÅÎÉÅ.
e
−x
0
= e
−x
· (−x)
0
= e
−x
· (−1) = −e
−x
.
ðÒÉÍÅÒ 17. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ (tg
√
x)
3
.
òÅÛÅÎÉÅ.
(tg
√
x)
3
0
= 3(tg
√
x)
2
(tg
√
x)
0
= 3(tg
√
x)
2
1
cos
2
√
x
(
√
x)
0
=
= 3(tg
√
x)
2
1
cos
2
√
x
1
2
√
x
=
3 tg
2
√
x
2
√
x cos
2
√
x
.
ðÒÉÍÅÒ 18. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ cos log
6
5x −log
6
cos 5.
òÅÛÅÎÉÅ.
(cos log
6
5x − log
6
cos 5)
0
= (cos log
6
5x)
0
− (log
6
cos 5)
0
= (cos log
6
5x)
0
=
= −sin log
6
5x ·(log
6
5x)
0
= −sin log
6
5x ·
1
(5x) ln 6
· (5x)
0
=
= −sin log
6
5x ·
1
5x ln 6
· 5 = −
sin log
6
5x
x ln 6
.
úÁÍÅÞÁÎÉÅ. ÷ÙÒÁÖÅÎÉÅ log
6
cos 5 ÉÚ ÐÒÉÍÅÒÁ 18 Ñ×ÌÑÅÔÓÑ ÞÉÓÌÏÍ, ÐÏ-
ÜÔÏÍÕ (log
6
cos 5)
0
= 0.
ðÒÉÍÅÒ 19. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ arctg
2
e
−x
.
§1. ðÒÏÉÚ×ÏÄÎÁÑ ÆÕÎËÃÉÉ                                                                   7

ÏÔËÕÄÁ
                                 0
                             x2                                 2                2
                 yx0   = e             = yu0 · u0x = eu · 2x = ex · 2x = 2xex .

  þÁÓÔÏ ÂÏÌÅÅ ÕÄÏÂÎÏ ÎÅÐÏÓÒÅÄÓÔ×ÅÎÎÏ ÎÁÈÏÄÉÔØ ÐÒÏÉÚ×ÏÄÎÙÅ ÐÒÏÍÅÖÕÔÏÞ-
ÎÙÈ ÆÕÎËÃÉÊ.
  ðÒÉÍÅÒ 14. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ ln sin x (ÓÍ. ÐÒÉÍÅÒ 12).
  òÅÛÅÎÉÅ.
                                           1                1
                   (ln sin x)0 =               (sin x)0 =       cos x = ctg x.
                                         sin x            sin x
                                                                         2
  ðÒÉÍÅÒ 15. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ y(x) = ex (ÓÍ. ÐÒÉÍÅÒ 13).
  òÅÛÅÎÉÅ.
                     2 0   2           2           2
             y (x) = ex = ex · (x2)0 = ex · 2x = 2xex .
              0



  ðÒÉÍÅÒ 16. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ e−x .
  òÅÛÅÎÉÅ.
                  0
               e−x = e−x · (−x)0 = e−x · (−1) = −e−x .
                                              √
  ðÒÉÍÅÒ 17. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ (tg x)3.
  òÅÛÅÎÉÅ.
        √      0     √      √          √                  1     √ 0
  (tg       x)3 = 3(tg x)2(tg x)0 = 3(tg x)2                √  (  x) =
                                                        cos2 x
                                                                            √
                                                      √ 2 1        1   3 tg2 x
                                                = 3(tg x)     √ √ = √         √ .
                                                          cos2 x 2 x 2 x cos2 x
  ðÒÉÍÅÒ 18. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ cos log6 5x − log6 cos 5.
  òÅÛÅÎÉÅ.

 (cos log6 5x − log6 cos 5)0 = (cos log6 5x)0 − (log6 cos 5)0 = (cos log6 5x)0 =
                                                              1
           = − sin log6 5x · (log6 5x)0 = − sin log6 5x ·           · (5x)0 =
                                                          (5x) ln 6
                                                            1             sin log6 5x
                                        = − sin log6 5x ·         ·5=−                .
                                                          5x ln 6            x ln 6
   úÁÍÅÞÁÎÉÅ. ÷ÙÒÁÖÅÎÉÅ log6 cos 5 ÉÚ ÐÒÉÍÅÒÁ 18 Ñ×ÌÑÅÔÓÑ ÞÉÓÌÏÍ, ÐÏ-
ÜÔÏÍÕ (log6 cos 5)0 = 0.
   ðÒÉÍÅÒ 19. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ arctg2 e−x .

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