y = f(x), (a,b). x (a,b), D x¹ 0, x+D x Î (a,b). M,P - f(x), x, x+D x (.21). M P M(x,f(x)), P(x+D x,f(x+D x). , M, P f(x) . MP X f (D x).
3. MP N M D x 0), f(x) M .
, f(x) M , limD x 0f (D x) = f 0, , OX.
:
1. f(x) x , f(x)
M(x,f(x)), f'(x).
: f'(x 0 ) , y = f(x) x 0, tg .
f(x) x0
y = f (x 0) +f' (x 0)(x-x 0)
3. y = 2x2-x+5 x = -0,5.
. x = -0,5
y' = 4 x- 1, y' (-0,5) = -3.
:
y = 6-3(x+ 0,5) y = -3 x+ 4,5.
h4> (a,b).
4 ( ). f(x) x, D y x
D y =A D x + a (D x) D x, | (1) |
A - , D x, limD x 0 a (D x) = 0.
, a(0) = 0. a(x) D x = 0. 1 , a (D x), D x - D x = 0 ,
D y =A D x +o (D x). | (2) |
1. x, , .
. . , (1). (1) D x¹ 0
D y/ D x = A+ a(D x).
D x 0, , A=f'(x).
. f'(x),
limD x 0D y/ D x = f' (x).
a(D x) = D y/ D x-f'(x). (1).
4. , |x| x = 0.
|
|
. x = 0:
D y = | D x|
limD x -0D y/ D x = -1, limD x +0D y/ D x = 1,
, |x| x = 0 .
.
2 ( ).
x, .
. x, (1), , limD x 0D y = 0, .
, . 4.
. X, . ( ), .
:
- , c' = 0.
- ,
u (x) v (x)) ' = u' (x) v' (x).
3. ,
(u (x) v (x)) ' = u' (x) v (x) +u (x) v' (x).
1. :
(cu (x)) ' = cu' (x).
4.
(u (x) /v (x)) ' = (u' (x) v (x) -u (x) v' (x)) /v 2(x)
, v(x)¹ 0.