. = f() , :
′ =.
.
y′ , f′(x).
( ) :
1) (Cu)′ = Cu′;
2) (u+v-w)′ = u′+v′-w′;
3) (uv)′ = u′v+uv′;
4)
:
1) ()′ = 0, ( = const);
2) ()′ = 1;
3)
4);
5);
6);
7) ()′ =;
8);
9);
10) ()′ =;
11) ()′ = -;
12) (tg x)′ =;
13) (ctg x)′ = -;
14) (arctg x)′ =;
15) (arcctg x)′ = -;
16) (arcsin x)′ =;
17) (arccos x)′ = -.
:
:
1) = 45 + - 8 -16;
= 45 + - 8-16 = 45 + -6 - 8- 16;
′ = 45-1 + (-6-6-1) - 8= 204 -7 - 6= 204 - -;
2) = (3 2 11);
′ = (3 2 11)′+ (3 2 -11)= (32 2)+
(3 - 2 11)= (32 -2) arctg x +;
3) =;
′ = = =
= =.
:
:
1)
2) y = 23 _ 52 + 7 - 12;
3) y = - 2 + 3 - 4;
4) y =2 - + 8;
5) y = 6 + 5 - 7;
6) y = tgx ctgx;
7) y = + -;
8) y = (2 2 + 6);
9) y = (85 3 + 4);
10) y =;
11) y =;
12) y =(-;
13) =;
14) =;
15) = x2 ;
16) = arctg .
17) = arctg x;
18) =;
19) =;
20) =;
21) = 3 (- 4).
:
:
1)
2) = 4 46 + 95 + 27 2 42;
3) = - +;
4) = 15 - 3 + +;
5) = - - + 0,75;
6) =;
7) = (2 3 + 3)(2 +2 1);
8) =;
9) =;
10) =;
11) = (23 + 3) arccosx.
2
: .
= f(u) u = () , ( ) = f((x)) () () : ′=f′(()) ′().
:
1) (un)′ = nun-1 u;
2) (′ = -;
3) ()′ =;
4) ()′ =;
5) ()′ =;
6))′ =;
7))′ =;
8))′ = - u′;
9) (tg u)′ =
10) ctg u)′ = -;
11) (arcsin u)′ =;
12) (arccos u)′ = -;
13) (arctg u)′ =;
14) (arcctg u)′ = -.
:
=;
′ = - = - - (10 3) = -.
:
:
1)
2) =;
3) =;
4) =;
5) =;
6) = ()2;
7) = (23 + 32 + 6 +1)4;
8) =;
9) = (7 +)3;
10) =);
11) =;
12) = (+ ()2)3;
13) =;
14) =;
15) =;
16) = ()5;
17) y = ()4;
|
|
|
18) y =;
19) y = (+))6;
20) y = -;
21) y=
22)
:
:
1)
2) = (2 3)5;
3) = ()2;
4) =;
5) =;
6) =;
7) =;
8) = (-)5;
9) =;
10) = + +;
11) =.
3
: , .
=.
() , F(;) = 0. , , , ′.
:
23 - + 3 = 0.
23 + 222 ′ - 2= 0;
222 ′ = 2- 23;
′ = =;
:, (t) (t)
, : ′() =.
:
′(t) =, ′(t) = 3 - 3 t2 = 3(1 t2);
′() = = 3(1 t2).
- =, u v , :
′ = vu′ +
:
=.
′ = (-) + =)+ +) = ( tg ).
:
:
1)
2) 2 + 5 + 2 7 = 0;
3) 2 + + = 0;
4) + 5 = 0;
5) 4 + 4 = 22;
6) 3 + = 0;
7)
8)
9)
10)
11)
12) =;
13) =;
14) =;
15) =;
16) =.
:
:
1)
2) 33 2 +3 = 0;
3) arctg = 0;
4) 2 + tg - = 0;
5) = ctg t, =;
6)
7) =;
8) =;
9) =;
10) =
4
