 .  .
 


              .      ,         .       .       ,    ,     .





 

 .  .



:

1)     .

2)  .

3)     .

4)      .

5)        .

6)        .

7)       .

8)     .

9)   .

10)   .

11)   .

12)     .

13)     .

14)   .

15)         .

16)   .

17)  - .

.

1))     .

     .

    ( )    ( ).

     .

   ,   - .

    S.

     ,  - .

    t.

     - ,    ().

 t ()   .

 S (t) (     )   .

     .

      .

       .    .

      .   .








     .       .

 S(t)     .

     .       .

,            -     .

    , ,    .

   ,     .

         .    ;   .

     .

    ,    ,      .








      ,       ?t.

? -  ().

? t -  .

     t0 ( ).

     :

v = S / t

 ,     -        .

     S0  ( ).

S1 - S0; S2 - S1;  S6 - S5

   ()     .

    ,    ?t.

    ,        ().

,    :

v = ? S / ? t



v = (S2 - S1) / (t2 - t1)

     S1  S2,      S (t1)  S(t2).

       t1  t2,     ? t.

      ? t,    .








      ,     :

v  = (S (?t + t) - S (t)) / ? t

       ,      ;     .

   ,    .

 t    (?t -> 0),      .

.

     t ( )  ,     ,  ? t -> 0.








  v (t)    S(t)   S ' (t)  




v (t) = S ' (t)








    .

    .

  v (t) -       .

      .

S (t) -  .

t  -    ().

S ' (t) = v (t) -    .

 ,    .        .

      .

     ;     (, ,    ),   S  t    f  x.

2))  .

    ,        ,  ,      . (?x -> 0)















? f = f (x1 + ?x) - f (x1) -  .

y = f (x) -  .

y ' = f ' (x) -    .

 ,    (, ,    ()).

    ,   .

     ,   .

     ,        (    ).








  x1  ,   f ' (x1) > 0 ( ).

   x4  ,   f ' (x4) < 0 ( ).

   x2; x3; x5      ,  f (x2) = 0; f (x3) = 0; f (x5) = 0 (  ).

3))     .

   .     .

     .

     ,          .

        .

,     ,   .

.

f (x) = 1/3 x ? - x

f ' (x) = x ? - 1

   .

     ,    .

    .

 ? - 1 = 0

x ? = 1

x1 = 1   x2 = - 1

:   1

x = 1; x = - 1 -   ,   ,      ,    .

 .

f (x) = 1/3 (x - 1) ? + 5

f ' (x) = (x - 1) ?

   .

     ,    .

     .

( - 1) ? = 0

x - 1 = 0        x - 1 = 0

x = 1              x = 1

: 1

    (  ).

x = 1 -   ,       ,      .

,      ,   .

   :

1)   ,        .       max.








2)   ,        .       min.








,      ;    ,    .

    .

   ,   .

f ' (x) < 0   .

   ,   .

f ' (x) > 0  .








 .

f (x) = 1/3 x ? - x - .

f  ' (x) = x ? - 1 -  .

        .

 ? - 1 = 0

x ? = 1

x = 1

x = - 1

:   1

            .

+   .

-   .








f ' (0) = 0 - 1 = - 1

     .

    x = 1  x = - 1    .

   .

   +  - ,   


 max.

x = - 1 (max).

  -  +,   


 min.

x = 1 (min).

    .








 .

f (x) = 1/3 (x - 1) ? + 5 - .

f ' (x) = (x - 1) ? -  .

       .

(x - 1) ? = 0

x - 1 = 0

x - 1 = 0

x = 1

 = 1

: x = 1

             .








f ' (0) = (0 - 1) ? = 1 ( +   ).

  x = 1  ,  .

     x = 1      ,     .

      .

    .








    .

     , :

1)         ,      .








2)         .

       .








      .








  y = f (x) ,      X     x0,  x0 ? X  f (x0) ? f (x)   x ? X.

   y = f (x) ,       X     x0,  x0 ? X  f (x0) ? f (x)   x ? X.

4))    .

1.     .

C -  ().

C ' = 0

2.   y = x  .

x ' = 1

3.      :

y = x ?

y ' = n x ???

(x ?) ' = n x ???

 1.

y = x ?

y ' = (x ?) ' = 2 x ??? = 2x

: 2x

 .

y = x ?

y ' = (x ?) ' = 3 x ??? = 3x ?

:  3x ?

 3.

y = ?x  y = x ???

y ' = (x ???) ' = 1/2 ? x ????? = 1/2 ? x ???? = 1/2 ? 1 / x ??? = 1 / 2?x

:  1 / 2?x.

 .

y = ??x = x ???

y ' = (x ???) ' = 1/3 ? x ????? = 1/3 ? x ???? = 1 / 3 x ??? = 1 / 3 ??x ?

: 1 / 3 ??x ?

 5.

y = 1 / x = x ??

y ' = (x ??) ' = - 1 x ???? = - x ?? = - 1 / x ?

: - 1 /  ?

 6.

y = 1 / ?x = x ????

y ' = (x ????) ' = - (1/2) x ?????? = - (1/2) x ???? = - 1 / 2x ??? = - 1 / 2?x ?

: - 1 / 2 ?x ?

 7.

y = 1 /  ? = x ??

y ' = (x ??) ' = - 2 x ???? = - 2 x ?? = - 2 /  ?

: - 2 / x ?

 8.

y = 1 / ??x = 1 / x ??? = x ????

y ' = (x ????) ' = - 1/3 ? x ?????? = - 1/ 3 ? x ???? = - 1 / 3x ??? = - 1 / 3 ??x ?

: - 1 / 3 ??x ?

 9

y = 1 / x ? =  x ??

y ' = (x ??) ' = - 3 x ???? = - 3 x ?? = - 3 / x ?

: - 3 / x ?

     .

1. (x ?) ' = 2x

2. (x ?) ' = 3x ?

3. (1/ ?x) ' = - 1/2 ? 1 / ?x ?

4. (??x) ' = 1 / (3 ??x ?)

5. (?x) ' = 1 / (2 ?x)

6. (1 / x) ' = - 1 / x ?

7. (1 / ??x) ' = - 1 / (3 ??x ?)

5))       .

1)     .

f (x) + g (x) = f ' (x) + g ' (x)

2)     .

f (x) - g (x) = f ' (x) - g ' (x)

3)           .

      .

(c f (x)) ' = c f ' (x)

 .

f (x) = - 3x ? + 2

f ' (x) = (- 3x ? + 2) ' = (- 3x ?) ' + (2) '= - 3 ? 2x + 0 = - 6

: - 6

 2.

y = x ? - x ?

y ' = (x ? - x ?) ' = (x ?) ' - (x ?) ' = 3x ? - 2x

:  3x ? - 2x

 .

y = 3x ? - 6x + 6

y ' = 3x ? 2x - 6 = 6x - 6

:  6x - 6

 .

f (x) = - 3 ? + 2x ? - x - 5

f ' (x) = - 3 ? 3x ? + 2 ? 2x - 1 - 0 = - 9x ? + 4x - 1

: - 9x ? + 4x - 1

6))       .

1)   ,               .

(f (x) g (x)) ' = f ' (x) g (x) + f (x) g ' (x)

 f (x) = U; g (x) = V,

(U V) ' = U ' V + U V '

2)   ,          ,        ,   .

(U / V) ' = (U ' V - U V ') / V ?

3)   .

  ,         .




  .


   .

   ,     (https://www.litres.ru/book/vasilisa-volkonskaya/proizvodnye-funktsii-primery-resheniia-74571762/)  .

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