Дифференциальное исчисление. - 3 стр.

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§1. ðÒÏÉÚ×ÏÄÎÁÑ ÆÕÎËÃÉÉ 3
ôÁË ËÁË × ËÁÞÅÓÔ×Å x
0
ÍÏÖÎÏ ×ÚÑÔØ ÌÀÂÏÅ ÞÉÓÌÏ, ÔÏ ÄÌÑ ÌÀÂÏÇÏ ÞÉÓÌÁ x
×Ù×ÏÄÉÍ
y
0
(x) = (sin x)
0
= cos x.
îÁÐÒÉÍÅÒ, y
0
(
π
2
) = cos
π
2
= 0.
1.2. ðÒÏÉÚ×ÏÄÎÙÅ ÏÓÎÏ×ÎÙÈ ÜÌÅÍÅÎÔÁÒÎÙÈ ÆÕÎËÃÉÊ
ðÒÉ×ÅÄ¾Í ÐÒÏÉÚ×ÏÄÎÙÅ ÏÓÎÏ×ÎÙÈ ÜÌÅÍÅÎÔÁÒÎÙÈ ÆÕÎËÃÉÊ.
(c)
0
= 0 (c ¡ ÞÉÓÌÏ);
(x
α
)
0
= αx
α−1
;
(e
x
)
0
= e
x
;
(a
x
)
0
= a
x
ln a (a > 0, a 6= 1);
(ln x)
0
=
1
x
;
(log
a
x)
0
x =
1
x ln a
(a > 0, a 6= 1, x > 0);
(sin x)
0
= cos x;
(cos x)
0
= −sin x;
(tg x)
0
=
1
cos
2
x
, x 6=
π
2
+ πn, n ∈ Z;
(ctg x)
0
= −
1
sin
2
x
, x 6= πn, n ∈ Z;
(arcsin x)
0
=
1
√
1−x
2
, |x| < 1;
(arccos x)
0
= −
1
√
1−x
2
, |x| < 1;
(arctg x)
0
=
1
1+x
2
;
(arcctg x)
0
= −
1
1+x
2
;
(sh x)
0
= ch x, ÇÄÅ sh x =
e
x
−e
−x
2
;
(ch x)
0
= sh x, ÇÄÅ ch x =
e
x
+e
−x
2
;
(th x)
0
=
1
ch
2
x
, ÇÄÅ th x =
sh x
ch x
;
(cth x)
0
= −
1
sh
2
x
, x 6= 0, ÇÄÅ th x =
sh x
ch x
.
ðÒÉÍÅÒ 3. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÙÅ ÓÌÅÄÕÀÝÉÈ ×ÙÒÁÖÅÎÉÊ: 5, ln tg
3
7
, 6
x
,
log
3
x.
òÅÛÅÎÉÅ. ðÒÏÉÚ×ÏÄÎÁÑ ÞÉÓÌÁ ÒÁ×ÎÁ ÎÕÌÀ, ÐÏÜÔÏÍÕ 5
0
= 0,
ln tg
3
7
0
= 0,
ÔÁË ËÁË 5 É ln tg
3
7
¡ ÞÉÓÌÁ.
äÌÑ ÎÁÈÏÖÄÅÎÉÑ ÐÒÏÉÚ×ÏÄÎÙÈ ÆÕÎËÃÉÊ 6
x
É log
3
x ×ÏÓÐÏÌØÚÕÅÍÓÑ ÔÁÂÌÉÞ-
ÎÙÍÉ ÆÏÒÍÕÌÁÍÉ ÄÌÑ ÐÒÏÉÚ×ÏÄÎÙÈ ÐÏËÁÚÁÔÅÌØÎÏÊ (ÐÒÉ a = 6) É ÌÏÇÁÒÉÆ-
ÍÉÞÅÓËÏÊ (ÐÒÉ a = 3) ÆÕÎËÃÉÊ, ÉÍÅÅÍ: (6
x
)
0
= 6
x
ln 6, (log
3
x)
0
=
1
x ln 3
.
ðÒÉÍÅÒ 4. ÷ÙÞÉÓÌÉÔØ ÐÒÏÉÚ×ÏÄÎÙÅ ÓÌÅÄÕÀÝÉÈ ÆÕÎËÃÉÊ: x
17
, x
π−e
,
1
x
,
√
x,
3
√
x
2
,
1
5
√
x
7
.
§1. ðÒÏÉÚ×ÏÄÎÁÑ ÆÕÎËÃÉÉ                                                      3

ôÁË ËÁË × ËÁÞÅÓÔ×Å x0 ÍÏÖÎÏ ×ÚÑÔØ ÌÀÂÏÅ ÞÉÓÌÏ, ÔÏ ÄÌÑ ÌÀÂÏÇÏ ÞÉÓÌÁ x
×Ù×ÏÄÉÍ
                                y 0 (x) = (sin x)0 = cos x.

îÁÐÒÉÍÅÒ, y 0 ( π2 ) = cos π2 = 0.


1.2. ðÒÏÉÚ×ÏÄÎÙÅ ÏÓÎÏ×ÎÙÈ ÜÌÅÍÅÎÔÁÒÎÙÈ ÆÕÎËÃÉÊ

    ðÒÉ×ÅÄ¾Í ÐÒÏÉÚ×ÏÄÎÙÅ ÏÓÎÏ×ÎÙÈ ÜÌÅÍÅÎÔÁÒÎÙÈ ÆÕÎËÃÉÊ.
   0
(c) = 0 (c ¡ ÞÉÓÌÏ);
(xα )0 = αxα−1;
(ex )0 = ex ;
(ax )0 = ax ln a (a > 0, a 6= 1);
(ln x)0 = x1 ;
(loga x)0x = x ln1 a (a > 0, a 6= 1, x > 0);
(sin x)0 = cos x;
(cos x)0 = − sin x;
(tg x)0 = cos12 x , x 6= π2 + πn, n ∈ Z;
(ctg x)0 = − sin12 x , x 6= πn, n ∈ Z;
                    1
(arcsin x)0 = √1−x     2 , |x| < 1;
                       1
(arccos x)0 = − √1−x       2 , |x| < 1;
                  1
(arctg x)0 = 1+x    2;
                      1
(arcctg x)0 = − 1+x      2;
                                    x      −x
(sh x)0 = ch x, ÇÄÅ sh x = e −e         2     ;
                                     x     −x
                                   e   +e
(ch x)0 = sh x, ÇÄÅ ch x = 2 ;
(th x)0 = ch12 x , ÇÄÅ th x = ch   sh x
                                       x
                                         ;
(cth x)0 = − sh12 x , x 6= 0, ÇÄÅ th x = ch     sh x
                                                   x
                                                     .
    ðÒÉÍÅÒ 3. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÙÅ ÓÌÅÄÕÀÝÉÈ ×ÙÒÁÖÅÎÉÊ: 5, ln tg 73 , 6x ,
log3 x.
                                                                             0
    òÅÛÅÎÉÅ. ðÒÏÉÚ×ÏÄÎÁÑ ÞÉÓÌÁ ÒÁ×ÎÁ ÎÕÌÀ, ÐÏÜÔÏÍÕ 50 = 0, ln tg 37 = 0,
ÔÁË ËÁË 5 É ln tg 73 ¡ ÞÉÓÌÁ.
    äÌÑ ÎÁÈÏÖÄÅÎÉÑ ÐÒÏÉÚ×ÏÄÎÙÈ ÆÕÎËÃÉÊ 6x É log3 x ×ÏÓÐÏÌØÚÕÅÍÓÑ ÔÁÂÌÉÞ-
ÎÙÍÉ ÆÏÒÍÕÌÁÍÉ ÄÌÑ ÐÒÏÉÚ×ÏÄÎÙÈ ÐÏËÁÚÁÔÅÌØÎÏÊ (ÐÒÉ a = 6) É ÌÏÇÁÒÉÆ-
ÍÉÞÅÓËÏÊ (ÐÒÉ a = 3) ÆÕÎËÃÉÊ, ÉÍÅÅÍ: (6x )0 = 6x ln 6, (log3 x)0 = x ln1 3 .
    ðÒÉÍÅÒ
      √          4. ÷ÙÞÉÓÌÉÔØ ÐÒÏÉÚ×ÏÄÎÙÅ ÓÌÅÄÕÀÝÉÈ ÆÕÎËÃÉÊ: x17, xπ−e, x1 ,
√ 3 2 1
  x, x , √5 x7 .

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