y = f (x). 1. D D = 1 , (. 1 = +D ).
. D , 1 > , , 1 < .
y = f (x) y 1= f (x 1).
D f (x)
D f (x) = f (x 1) - f (x) = y 1 y D f (x) = f ( + D x) f (x).
D 0 , f (x) , f (x)
(2.1)
- , f (x). .
. f (x), 1 = + D , = D , NC = D f (x).
(2.2)
MN (. .2.1).
D 0, N M, MN f (x) M, α φ. (2.1) (2.2) , f ¢(x) y = f (x) (, f (x)).
,
( )
,
. 2.1.
.
.
1.
(C)` = 0 (2.3)
2. f 1 (x) f 2(x)
() = 1f1 (x) +c2f2 (x), (2.4)
1 c2 ,
¢(x) = (1f1 (x) +c2f2 (x)) ¢ = 1f1 ¢ (x) +c2f2 ¢ (x). (2.5)
, D (x).
1. (x 1) (x) .
D (x) = (x 1) - (x) = (1 f 1(x 1) + 2 f 2(x 1)) - (1 f 1(x) + 2 f 2(x)).
f 1 (x) f 2(x) 1 2. f 1 (x) f 2(x)
|
|
D (x) = (1 f 1(x 1) - 1 f 1(x)) + (2 f 2(x 1) - 2 f 2(x)) = 1 (f 1(x 1) - f 1(x)) +
(2.6)
+ 2 (f 2(x 1) - f 2(x))= 1 D f 1(x) + 2 D f 2(x 1).
D (x) (2.6) (2.1) ( ) :
,
.
.
( (x)) ¢= ¢(x).
3. (x) = f (x) g (x) :
(x) = (f (x)g(x))¢ = f ¢(x) g (x) + f (x) g¢(x). (2.7)
n
(f 1(x) f 2(x) .. . f n(x))¢ =
= f 1(x)¢ f 2(x) . f n(x)+ f 1(x) f 2(x)¢ . f n(x)+.+ f 1(x) f 2(x) .. f n(x)¢
4.
(2.8)
.
1. (6 sin x - 2 ln x)¢ = (6 sin x)¢ - (2 ln x)¢ = 6 (sin x)¢ - 2 (ln x)¢ = 6 cos x -
. 1 . . 2 .
. g (x) f (x). (x) = f (g (x)) f g.
. D (x). 1 = x + D x. g (x + D x) g (x)
D g (x) = g (x + D x) - g (x) g (x + D x) = g (x) + D g (x).
f (g (x + D x)) f (g (x)).
D f = f (g (x +D x)) f (g (x)) = f (g (x) + D g (x)) f (g (x)). (2.9)
(2.9) (2.1). D g (x) .
(2.10)
2. (lnx∙cosx)' = ∙cosx - lnx∙sinx.
3.
1.
: , . |
: , , |
: . |
: , . |
: . |
: . |
: . |
:; |
: ; . |
: . |
: . |
:, ; |
:; . |
(2.11)
: = ln (sin (x 2)). : f = ln g, g = sinh, h = x 2.
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2.
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: |
. , :
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1.
2. , .
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.
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.
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