y = f(x) X. y = f(x) o
= .
, f(x) xo; .
¥ ( - ¥), , o , , f(x) o .
y ¢, f ¢(xo), , .
. , y=f(x) o; - , s = s(t) to.
- , u = u(x), v = v(x) - , :
1) () ' = 0, (cu) ' = cu';
2) (u+v)' = u'+v';
3) (uv)' = u'v+v'u;
4) (u/v)' = (u'v-v'u)/v2;
5) y = f(u), u = j(x), .. y = f(j(x)) - , , j f, ,
;
6) y = f(x) x = g(y), ¹ 0, .
.
1. (um)' = m um-1 u' (m Î R).
2. (au)' = au lna× u'.
3. (eu)' = eu u'.
4. (loga u)' = u'/(u ln a).
5. (ln u)' = u'/u.
6. (sin u)' = cos u× u'.
7. (cos u)' = - sin u× u'.
8. (tg u)' = 1/ cos2u× u'.
9. (ctg u)' = - u' / sin2u.
10. (arcsin u)' = u' / .
11. (arccos u)' = - u' / .
12. (arctg u)' = u'/(1 + u2).
13. (arcctg u)' = - u'/(1 + u2).
-
y=uv, (u>0), u v , u', v'.
y=u v, ln y = v ln u.
3, 5 , :
y'/y = vu'/u +v' ln u, y' = y (vu'/u +v' ln u).
,
(u v)'=u v (vu'/u+v' ln u), u > 0.
, y = x sin x, y' = x sin x (sin x/x + cos x× ln x).
y = f(x) x, .. y', = y'+a, a0 D 0; D y = y' D + a x.
, D, dy: dy = y' D. y=x, dx = x'D = 1×D =D, dy=y'dx, . . .
D y , d y .
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y=f(x) y ¢= f ¢(x). f(x), , .
:
- ,
-
n- - .
15. y=(3x3-2x+1)×sin x.
. 3, y'=(3x3-2x+1)'×sin x + (3x3-2x+1)×(sin x)' =
= (9x2-2)sin x + (3x3-2x+1)cos x.
16. y', y = tg x + .
. , : y'=(tgx + )' = (tgx)' + ()' = + = .
17. y= ,
u=x4 +1.
. , : y'x =y 'u u'x =()'u(x4 +1)'x =(2u + . u=x4 +1,
(2 x4 +2+ .
18. y= .
. y= : y = eu u = x2. : y'x =y 'u u'x = (eu)'u(x2)'x = eu ×2x. x2 u, y=2x .
19. y=ln sin x.
. u=sin x, y=ln u y' = (ln u)'u(sin x)'x= .
20. y= .
. , , 5:
.
21. y=ln .
. , :
y=5/3ln(x2+4) +7/3ln(3x-1)-2/3ln(6x3+1)-1/3tg 5x.
, :
.
2.2. .
. , f(x) , - x, f '(x) ; g(x) , . . g(x) x, g'(x) .
- , .. . , , . . . , , , , , , .
v x : v = f(x) - v/x, - .
.
u = u(t), u t.
Dt = t1 - t0: Du = u(t1) - u(t0) = u(t0+Dt) - u(t0). , .. z .= Du/Dt.
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z(t0) t0 , z . Dt0: . , , : z(t0) = u'(t0).
K x. K = K(x). , D . x+ D K(x + D). , D DK = K(x + D) - K(x).
DK/D. .
.
u(x) x , .
, . ( ) , . ( ). , y = f(x), y ¢ = f ¢(x). y = f(x) x
.
Ex (y) = x/y f ¢ (x) = .
x ( ), 1%. , , , . , . . , , . . , . . , - , .
y=f(x) () , x1< x2 f(x1) < f (x2) (f(x1) > f(x2)).
y = f(x) [a, b] (), f ¢(x) > 0 (f ¢(x) < 0).
x () f(x), x, f(x) £ f(x) (f(x) ³ f(x)).
, - .
. x f(x), f ¢(x) = 0, f ¢(x) . , . .
. x - . f ¢ (x) x , x , - . , x .
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. f(x)
f ¢ (x) x x. f ¢(x) = 0, >0 ( <0), x () f(x). =0, , .
[a,b] y = f(x) , [a,b].
22. f(x) = 2x3 - 15x2+ 36x - 14.
. f ¢ (x) = 6x2 - 30x +36 = 6(x -2)(x - 3), x1 = 2 x2 = 3. . x1 = 2 , . x2 = 3 , x2 = 3 .
x1 = 2 x2 = 3, : f(2) = 14 f(3) = 13.
23. , , . a . ?
. x y. S = xy. y - , . 2x + y = a. y = a - 2x S = x(a - 2x), 0 £ x £ a/2 ( ). S ¢ = a - 4x, a - 4x = 0 x = a/4,
y = a - 2×a/4 =a/2. x = a/4 - , , . x < a/4 S ¢ >0, x >a/4 S ¢ <0, , x=a/4 S . S(a/4) = a/4(a - a/2) = a2 /8 (. ).
S [0, a/2] S(0) S(a/2) , . , y = 2x.
24. V=16p 50 3. ( R ), ?
. S = 2pR(R+). V = pR2 Þ = V/pR2 =16p/ pR2 = 16/ R2. , S(R) = 2p(R2+16/R). :
S ¢(R) = 2p(2R- 16/R2) = 4p (R- 8/R2). S ¢(R) = 0 R3 = 8, ,
R = 2, = 16/4 = 4.