. .
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 = 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 .
y=f(x) y ¢= f ¢(x). f(x), , .
:
- ,
|
|
-
n- - .
1. 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.
2. y= .
. y= : y = eu u = x2. : y'x =y 'u u'x = (eu)'u(x2)'x = eu ×2x. x2 u, y=2x .
- , .
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 .
. f(x) f ¢ (x) x x. f ¢(x) = 0, >0 ( <0), x () f(x). =0, , .
[a,b] y = f(x) , [a,b].
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.