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y=f(x), x. Δx - x. Δy Δf , f(x+Δx) f(x). , x, Δx Δf.

Δf /Δx, 3.1, α, MN y = f(x) c .

 

 

. 3.1.

 

, Δx, , . N y = f(x), M, MN M , Δx α φ x. , , y = f(x) x, .

Δy / Δx , (f(x + Δx) f(x)) / Δx, x Δx. Δx = 0. .

(f(x + Δx) f(x)) / Δx Δx = 0, y = f(x) x y′ f′(x): .

y = f(x) .

x (a, b) f′(x), f(x) (a, b ).

, f(x) x .

- x. , f′ (x) ≈ Δf / Δx, , Δx. f′ (x) Δf Δx.

f(x) , . x0, . . 3.2.

 

 

. 3.2.

 

y = ⎢x ⎢ x = 0, .

.

1. , .

2. f′ (x), ‑ , CF(x) : (Cf(x))′ = Cf′ (x).

3. f′ (x) g′ (x), S(x) = f(x) + g(x) : S′ (x) = f′ (x) + g′ (x).

4. f′ (x) g′ (x), P(x) = f(x)g(x) : P′ (x) = f′ (x)g(x) + f(x)g′ (x).

5. f′ (x) g′ (x) g(x) ≠ 0, D(x) = f(x) / g(x) : D′ (x) = (f′ (x) g(x) - f(x) g′ (x)) / g2(x).

. .

g(x) x, f(z) z = g(x). F(x) = f(g(x) x F′ (x) = f′ (z) g′ (x).

.

 

 

: y1 = f1(x) y2 = f2(x), f1′ (x) f2′ (x) D. - x D Δx. Δy1 = f1(x + Δx) - f1(x) Δy2 = f2(x + Δx) - f2(x). , 3, , Δy1 Δy2 : Δy1 = (C1 - A1) + (B1 - C1); Δy2 = (C2 - A2) + (B2 - C2) (1)

 

 

. 3.3.

 

(1) : C1 A1 = tgα1 Δx = f1′ (x)Δx; C2 A2 = tgα2 Δx = f2′ (x)Δx.

f′ (x) Δx y = f(x) x. ( , x ). Δx ( Δx). , Δx k , k .

(1) :

Δy1 = f1′ Δx + r1; Δy2 = f2′ Δx + r2. (2)

r1 = B1 C1; r2= B2 C2.

r1 r2 (2) Δx k k , , 3 4, , r1 r2 , Δx.

 

 

. 3.4.

 

β (z) z = z0, .

β (z) γ (z) z = z0.. β (z) , γ (z), .

r1 r2 (2) Δx, Δx = 0. , . , r1(Δx) r2(Δx) , Δx, Δx = 0.

y = f(x) , , Δy = f′(x) Δx +β (Δx), β (Δx) ‑ , Δx, Δx = 0.

, Δx, y = f(x), f′ (x) Δx, dy: dy = f′ (x) Δx. (3)

f(x) = x, , x′ = 1, (3) : dx = Δx. y = x, , . , y = x . (3) dy = f′ (x) dx. , ,

f(x) x.

:

1. dC = 0 ( C -);

2. d(Cf(x)) = Cdf(x);

3. df(x) dg(x), d(f(x) + g(x)) = df(x) + dg(x), d(f(x)g(x)) = g(x)df(x) + f(x)dg(x). g(x) ≠0,

y = f(x) ‑ , x, dy = df(x) = f′ (x)dx. x x(t) t, y = F(t) = f(x(t)) - t, dy = F′(t)dt = f′ (x)x′ (t)dt. x′ (t)dt = dx : dy = f′ (x)dx.

y=f(x) , Δx = dx , f(x) . .

 

f ′(x), , (f′(x))′. f(x) f′′(x). f , t f′(t), f′′(t).

, . , f(x) f′′′(x).

n- f (n)(x) , (n+1)- f(x): f (n + 1)(x) = (f(n)(x))′.

, , .

 



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