2
2001
.
, .
. f (x) = 0 , .
f(x)
f(x0 +Dx) P
Df
f(x0) M
a b Dx
0 x0 x0 + Dx x
f (x) (a, b). .
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a - f (x) (x0, f(x0)).
, - .
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f(t), t- , f(t)- ( ) .
, - , .. .
.
. () f (x) = 0 () , .
f (x) = 0, . , . - 0, - , 0, .
: f(x) = ïxï- = 0 , , , .
. ( ) f (x) 0, .
, .
.
f (x) = u, g (x) = v - , .
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1) (u v) ¢ = u ¢ v ¢
2) (u × v) ¢ = u × v ¢ + u ¢ × v
3) , v ¹ 0
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1)¢ = 0; 9)
2)(xm)¢ = mxm-1; 10)
3) 11)
4) 12)
5) 13)
6) 14)
7) 15)
8) 16)
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. y = f (x); u = g (x), u f.
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( , Dx0, Du0, .. u = g(x) )
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(lnïxï)¢= , .. .
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f(x).
, ,
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- .
, , , . , .
u = f(x) v = g(x) , , f(x)>0.
y = uv. , :
lny = vlnu
. .
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= f (x) , x = g (y) , .
x = g(y) :
.. g¢(y) ¹ 0
.. .
. arctg.
arctg , tg, .. :
,
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.. :
, , .
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y = f(x) :
: , a0, D0.
: .
aDx- , f¢(x)Dx, .. f¢(x)Dx- D.
. f(x) .
dy df(x).
, dy = f¢(x)Dx
dy = f ¢ (x) dx.
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y
f(x)
K
dy
M Dy
L
a
x x + Dx x
DMKL: KL = dy = tg a × D x = y ¢ × D x
, f (x) .
.
u = f (x) v = g (x)- , , :
1) d(u v) = (u v) ¢ dx = u ¢ dx v ¢ dx = du dv
2) d(uv) = (uv) ¢ dx = (u ¢ v + v ¢ u)dx = vdu + udv
3) d(Cu) = Cdu
4)
.
.
y = f(x), x = g(t), .. - .
ࠠ dy = f ¢(x) g ¢(t) dt = f ¢(x) dx.
, dy , - , .
, - ,
dx = Dx,
t, D ¹ dx.
, dy = f¢(x)Dx .
. .
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