, , . .
f (x) x 0, :
1) f (x) x 0, f (x 0);
2) ;
3) = f (x 0).
.
1) , 2) 3) 4) = 0.
, 1) 3),
= = f (x 0) f (x 0) = 0.
, f (x 0) = 0,
= = + = f (x 0), , f (x 0).
D x = x x 0. x = x 0 + D x, f (x) = f (x 0 + D x) = 0 = 0. x 0 x, f (x) x: = 0. D x , D y = f (x + D x) f (x) f (x) x. , x f (x) .
1. f (x) = C.
1) x 0 .
4) = .
f (x) = C x 0 .
2. f (x) = .
1) x 0 , x 0 = 0.
4) = .
f (x) = x 0 ¹ 0 .
3. f (x) =
, ( ¥; 0) (0; +¥) f (x) , , x ¹ 0.
x = 0 f (x) : f (0) = 0, ,
, . , x = 0 f (x) .
2. ,
1. .
f (x) g (x) x 0, f (x) + g (x), f (x) g (x), f (x) g (x), g (x 0) ¹ 0, x 0.
.
. f (x) g (x) x 0, x 0 , x 0. , f (x) + g (x) x 0, f (x 0) + g (x 0). f (x) + g (x) x0 , = + . f (x) g (x) x 0, = + = f (x 0) + g (x 0), f (x) + g (x) x 0.
, .
2. .
u (x) x 0, f (u) u 0, u 0 = u (x 0). f (u (x)) x 0.
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.
.
y = f (x), D E, E x = g (y) , x 0 Î D y 0 Î E, y 0 = f (x 0), x 0 = g (y 0). g f f 1. , y x f 1(y) y , x 1 ¹ x 2 y 1 = f (x 1) y 2 = f (x 2) f : y 1 ¹ y 2. , f (x) .
3. .
f (x) f 1(y) x 0, f 1(y) y 0 = f (x 0).
.
. f (x) f 1(y) x, D y f (x) D x.
, D y = f (x + D x) f (x), D x = f 1(y + D y) f 1(y) , , .
4. .
.
.
y = x, y = sin x y = e x.
y = x , y = x x.
y = kx + b x.
y = sin x
= = 1 .
, Dx , y = . , y = sin x x.
y = cos x = . . y = arcsin x, y = arcos x, y = arctg x, y = arcctg x .
y = e x e x 1 0 = 0. , y = e x x.
y = ln x (x > 0) y = e x.
y = a x y = a x = . , y = a x x .
y = log a x y = log a x = . , y = log a x x .
y = x a a y = x a = . , y = x a x .
. . , .
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5. .
f (x) x0 f (x 0) > A, A , U (x 0,d) x 0, x f (x) > A.
.
f (x) x0, , f (x 0), f (x 0) > A, > A. (x 0,d), x f (x) > A. x 0, : U (x 0,d) = (x 0,d) È { x 0}. .
. f (x 0) < A, U (x 0,d), x f (x) < A. .