Íåïðåðûâíîñòü
Îïðåäåëåíèå
Ïóñòü f : D ⊂ R → R, a ∈ D. Ôóíêöèÿ f íàçûâàåòñÿ
òî÷êå a, åñëè
∀ε > 0 ∃δ > 0 ∀x ∈ D : |x − a| < δ
Ôóíêöèÿ f íàçûâàåòñÿ
íåïðåðûâíîé
Ôóíêöèÿ f íàçûâàåòñÿ
íåïðåðûâíîé
|f (x) − f (a)| < ε.
â òî÷êå a
∀ε > 0 ∃δ > 0 ∀x ∈ D : 0 ⩽ a − x < δ
ñëåâà
, åñëè
|f (x) − f (a)| < ε.
â òî÷êå a
∀ε > 0 ∃δ > 0 ∀x ∈ D : 0 ⩽ x − a < δ
íåïðåðûâíîé
, åñëè
ñïðàâà
|f (x) − f (a)| < ε.
â
Ïðèìåð 1. Äîêàçàòü, ÷òî ôóíêöèÿ f (x) = √1 − x íåïðåðûâíà â
êàæäîé òî÷êå a < 1 è íåïðåðûâíà ñëåâà â òî÷êå a = 1.
Ïóñòü a < 1. Âîçüì¼ì ε > 0. Íàì íóæíî íàéòè òàêîå δ > 0, ÷òîáû
äëÿ âñåõ x < 1 : |x − a| < δ âûïîëíÿëîñü íåðàâåíñòâî
√
√
1−x−
√
1−x−
√
1 − a < ε.
|a − x|
δ
|a − x|
√
<√
<√
1−x+ 1−a
1−a
1−a
√
⇒ δ := ε 1 − a
1−a = √
Äîêàæåì íåïðåðûâíîñòü ñëåâà â òî÷êå a = 1. Âîçüì¼ì ε > 0. Íàì
íóæíî íàéòè òàêîå δ > 0, ÷òîáû äëÿ âñåõ x < 1 : 0 ⩽ 1 − x < δ
âûïîëíÿëîñü íåðàâåíñòâî
√
| 1 − x − 0| < ε
⇔
ßñíî, ÷òî äîñòàòî÷íî âçÿòü δ := ε2 .
√
1 − x < ε.
Íàïîìèíàíèå 1. Ôóíêöèÿ íåïðåðûâíà â òî÷êå a òîãäà è òîëüêî
òîãäà, êîãäà ñóùåñòâóþò êîíå÷íûå f (a−), f (a+) è
f (a−) = f (a+) = f (a).
Íàïîìèíàíèå 2. Ôóíêöèÿ íàçûâàåòñÿ
ïîëó÷àåòñÿ èç ôóíêöèé
xα ,
ax ,
arcsin x,
ln x,
sin x,
arccos x,
, åñëè îíà
ýëåìåíòàðíîé
cos x,
arctg x,
tg x,
ctg x
arcctg x
ñ ïîìîùüþ êîíå÷íîãî ÷èñëà àðèôìåòè÷åñêèõ äåéñòâèé è îïåðàöèé
êîìïîçèöèè.
Âñå ýëåìåíòàðíûå ôóíêöèè íåïðåðûâíû íà ñâîèõ îáëàñòÿõ
îïðåäåëåíèÿ.
Ïðèìåð 2. Èññëåäóåì íà íåïðåðûâíîñòü ôóíêöèþ
f (x) = arctg
1
x−4
.
îïðåäåëåíà íà ìíîæåñòâå R \ {4} è ÿâëÿåòñÿ ýëåìåíòàðíîé
ìîæåò èìåòü ðàçðûâ òîëüêî â òî÷êå 4.
Âû÷èñëèì îäíîñòîðîííèå ïðåäåëû â ýòîé òî÷êå:
f
f
lim
x→4−
lim
1
= −∞
x−4
x→4+
1
= +∞
x−4
⇒
⇒
f (4−) = lim arctg
x→4−
1
π
=−
x−4
2
f (4+) = lim arctg
x→4+
1
π
=
x−4
2
â òî÷êå x = 4 ôóíêöèÿ f èìååò ðàçðûâ ïåðâîãî ðîäà; â
îñòàëüíûõ òî÷êàõ ôóíêöèÿ íåïðåðûâíà.
⇒
⇒
Ïðèìåð 3. Èññëåäóåì íà íåïðåðûâíîñòü ôóíêöèþ

1
2

 5 (2x + 3),
f (x) = 6 − 5x,


x − 3,
x ⩽ 1,
1 < x < 3,
x ⩾ 3.
Íà êàæäîì èç ïðîìåæóòêîâ (−∞, 1), (1, 3) è (3, +∞) f ÿâëÿåòñÿ
ýëåìåíòàðíîé ôóíêöèåé, è ïîòîìó íåïðåðûâíà
⇒ ðàçðûâû ìîãóò áûòü ëèøü â òî÷êàõ x = 1 è x = 3.
Âû÷èñëèì ñîîòâåòñòâóþùèå îäíîñòîðîííèå ïðåäåëû.
f (1−) = lim
x→1
1
(2x2 + 3)
5
=1
f (1+) = lim (6 − 5x) = 1
x→1
Ïîñêîëüêó f (1) = 1, ïîëó÷àåì, ÷òî f íåïðåðûâíà â òî÷êå 1.
f (3−) = lim (6 − 5x) = −9
x→3
⇒
x=3
f (3+) = lim (x − 3) = 0
x→3
ÿâëÿåòñÿ äëÿ f òî÷êîé ðàçðûâà ïåðâîãî ðîäà.
Ïðèìåð 4. Èññëåäóåì íà íåïðåðûâíîñòü ôóíêöèþ
.
Îíà îïðåäåëåíà ïðè x ̸= 8 è ÿâëÿåòñÿ ýëåìåíòàðíîé. Ïîýòîìó
äîñòàòî÷íî ðàññìàòðèâàòü ëèøü òî÷êó x = 8.
f (x) = 71/(x−8) + 2
lim
x→8−
lim
x→8+
1
= −∞
x−8
1
= +∞
x−8
=⇒
=⇒
f (8−) = 2,
f (8+) = +∞.
Òàêèì îáðàçîì, â òî÷êå x = 8 ó ôóíêöèè f ðàçðûâ âòîðîãî ðîäà.
Ïðîèçâîäíûå
Îïðåäåëåíèå (1)
Ïóñòü f : ⟨a, b⟩ → R, x0 ∈ ⟨a, b⟩. Åñëè ñóùåñòâóåò òàêîå ÷èñëî
, ÷òî
A∈R
f (x) = f (x0 ) + A(x − x0 ) + o(x − x0 ),
òî ôóíêöèÿ f íàçûâàåòñÿ
A
ôóíêöèè f â òî÷êå x0 .
äèôôåðåíöèðóåìîé
x → x0 ,
â òî÷êå x0 , à ÷èñëî
ïðîèçâîäíîé
Îïðåäåëåíèå (2)
Ïóñòü f : ⟨a, b⟩ → R, x0 ∈ ⟨a, b⟩. Åñëè ñóùåñòâóåò ïðåäåë
lim
x→x0
f (x) − f (x0 )
,
x − x0
ðàâíûé ÷èñëó A ∈ R, òî ôóíêöèÿ f íàçûâàåòñÿ
â òî÷êå x0 , à ÷èñëî A å¼
â òî÷êå x0 .
äèôôåðåíöèðóåìîé
ïðîèçâîäíîé
Äëÿ ïðîèçâîäíîé ôóíêöèè f â òî÷êå x0 èñïîëüçóþò îáîçíà÷åíèå
f ′ (x0 ).
Èòàê, äëÿ äèôôåðåíöèðóåìîé â òî÷êå x0 ôóíêöèè f
f ′ (x0 ) = lim
x→x0
Òàáëèöà ïðîèçâîäíûõ
f (x) − f (x0 )
.
x − x0
′
(const) = 0
′
(xα ) = αxα−1
â ÷àñòíîñòè, (ex )′ = ex
1
1
′
′
(loga x) =
, â ÷àñòíîñòè, (ln x) =
x ln a
x
′
(ax ) = ax ln a,
′
(sin x) = cos x
′
(cos x) = − sin x
1
cos2 x
1
′
(arcsin x) = √
1 − x2
1
′
(arctg x) =
1 + x2
′
(tg x) =
′
(sh x) = ch x
′
(th x) =
Íàïîìèíàíèå:
sh x =
1
ch2 x
′
(ctg x) = −
1
sin2 x
1
1 − x2
1
′
(arcctg x) = −
1 + x2
′
(arccos x) = − √
′
(ch x) = sh x
′
(cth x) = −
1
sh2 x
ex − e−x
2
ch x =
ex + e−x
2
sh x
ch x
cth x =
ch x
sh x
th x =
Ñâîéñòâà ïðîèçâîäíûõ è ïðàâèëà
äèôôåðåíöèðîâàíèÿ
Ëèíåéíîñòü äèôôåðåíöèðîâàíèÿ. Åñëè ôóíêöèè
äèôôåðåíöèðóåìû â òî÷êå x ∈ ⟨a, b⟩, α, β ∈ R, òî
äèôôåðåíöèðóåìà â òî÷êå x è
f, g : ⟨a, b⟩ → R
αf + βg
ôóíêöèÿ
′
(αf + βg) (x) = αf ′ (x) + βg ′ (x).
Èíûìè ñëîâàìè, ïðîèçâîäíàÿ ñóììû ðàâíà ñóììå ïðîèçâîäíûõ, à
ïîñòîÿííûé ìíîæèòåëü ìîæíî âûíîñèòü çà çíàê ïðîèçâîäíîé.
Íàïðèìåð,
=
3 5/3
2
x + x−2 +
5
x
′
=
′
′
3 5/3 ′
x
+ x−2 + 2 x−1 =
5
3 5 5/3−1
· x
− 2x−2−1 + 2 · (−1)x−1−1 = x2/3 − 2x−3 − 2x−2 .
5 3
Ïðîèçâîäíàÿ ïðîèçâåäåíèÿ. Åñëè ôóíêöèè f, g : ⟨a, b⟩ → R
äèôôåðåíöèðóåìû â òî÷êå x ∈ ⟨a, b⟩, òî ôóíêöèÿ f g
äèôôåðåíöèðóåìà â òî÷êå x è
(f g)′ (x) = f ′ (x)g(x) + f (x)g ′ (x).
Íàïðèìåð,
′
′
((x + 1) tg x) = (x + 1) tg x + (x + 1)(tg x)′ = tg x +
x+1
.
cos2 x
Äëÿ ïðîèçâîëüíîãî ÷èñëà ñîìíîæèòåëåé ýòî ñâîéñòâî âûãëÿäèò
òàê:
′
(f1 f2 · . . . · fn ) = f1′ f2 · . . . · fn + f1 f2′ · . . . · fn + . . . + f1 f2 · . . . · fn′ .
Íàïðèìåð,
′
′
′
′
(2x ln x arctg x) = (2x ) ln x arctg x+2x (ln x) arctg x+2x ln x (arctg x) =
= 2x ln 2 ln x arctg x + 2x ·
1
1
· arctg +2x ln x ·
.
x
1 + x2
Ïðîèçâîäíàÿ ÷àñòíîãî. Åñëè ôóíêöèè f, g : ⟨a, b⟩ → R
äèôôåðåíöèðóåìû â òî÷êå x ∈ ⟨a, b⟩ è g(x) ̸= 0, òî ôóíêöèÿ fg
äèôôåðåíöèðóåìà â òî÷êå x è
′
f
f ′ (x)g(x) − f (x)g ′ (x)
(x) =
.
g
g 2 (x)
Íàïðèìåð,
2x
arcsin x
′
′
′
2x ln 2 arcsin x −
(2x ) arcsin x − 2x (arcsin x)
=
=
arcsin2 x
arcsin2 x
x
√2
1−x2
.
Ïðîèçâîäíàÿ êîìïîçèöèè. Åñëè ôóíêöèÿ f : ⟨a, b⟩ → ⟨c, d⟩
äèôôåðåíöèðóåìà â òî÷êå x ∈ ⟨a, b⟩, à ôóíêöèÿ g : ⟨c, d⟩ → R
äèôôåðåíöèðóåìà â òî÷êå f (x), òî ôóíêöèÿ g ◦ f
äèôôåðåíöèðóåìà â òî÷êå x è
′
(g ◦ f ) (x) = g ′ (f (x)) f ′ (x).
Íàïðèìåð,
′
(sin(ln x)) = cos(ln x)(ln x)′ =
(çäåñü f (x) = ln x, g(x) = sin x).
cos(ln x)
x
Ïðèìåð 5.
′ ′
′
′
tg x
tg x
+ sin cos x2 + ln2 x = √
+ sin cos x2 + ln2 x
4
4
1+x
1+x
√
1+x4
1√ 1
4 ′
′
cos2 x − tg x · 2 1+x4 x
=
+cos cos x2 · cos x2 +2 ln x·(ln x)′ =
4
1+x
√
=
√
1+x4
cos2 x
− tg x ·
1+
1√ 1
2 1+x4
x4
√
=
1+x4
cos2 x
−
1+
· 4x3
3
2x
√ tg x
1+x4
x4
′
1
+cos cos x2 ·(− sin x2 )· x2 +2 ln x· =
x
2 ln x
− 2x cos cos x2 sin x2 +
x
Ïðèìåð 6.
(tg x)arctg x
′
′
′
= earctg x ln(tg x) = earctg x ln(tg x) · (arctg x ln(tg x)) =
′
= (tg x)arctg x · (arctg x ln(tg x)) =
′
′
= (tg x)arctg x · (arctg x) ln(tg x) + arctg x (ln(tg x))
ln(tg x) arctg x
′
arctg x
= (tg x)
+
· (tg x) =
1 + x2
tg x
ln(tg x)
arctg x
= (tg x)arctg x
+
1 + x2
tg x cos2 x
=
Ëîãàðèôìè÷åñêîå äèôôåðåíöèðîâàíèå
Ïî ïðàâèëó äèôôåðåíöèðîâàíèÿ ñëîæíîé ôóíêöèè
′
(ln f ) =
f′
.
f
Îòñþäà ñëåäóåò ôîðìóëà
′
f ′ = f (ln f ) .
ż ïðèìåíÿþò òîãäà, êîãäà ëîãàðèôì ôóíêöèè f ÿâëÿåòñÿ áîëåå
ïðîñòîé ôóíêöèåé, ÷åì ñàìà f .
Ïðèìåð 7. Âû÷èñëèì ïðîèçâîäíóþ ôóíêöèè
f (x) =
ln f (x) = ln
esin 5x
(3x − 2)2
esin 5x
.
(3x − 2)2
= ln esin 5x −ln (3x − 2)2 = sin 5x−2 ln(3x−2)
′
(sin 5x − 2 ln(3x − 2)) = 5 cos 5x −
esin 5x
(3x − 2)2
′
=
esin 5x
(3x − 2)2
6
3x − 2
5 cos 5x −
⇒
6
3x − 2
Ïðèìåðr 8. Âû÷èñëèì ïðîèçâîäíóþ ôóíêöèè
f (x) =
7
x2 + 3
arccos 4x.
x2 − 3
r
ln f (x) = ln
7
x2 + 3
arccos 4x
x2 − 3
!
=
1
ln(x2 + 3) − ln(x2 − 3) + ln arccos 4x
7
′
1
ln(x2 + 3) − ln(x2 − 3) + ln arccos 4x =
7
2x
2x
4
1
− 2
−√
⇒
=
2
7 x +3 x −3
1 − 16x2 arccos 4x
!′
r
2
7 x + 3
arccos 4x =
x2 − 3
=
r
=
7
x2 + 3
arccos 4x
x2 − 3
1
2x
2x
4
−
−√
.
7 x2 + 3 x2 − 3
1 − 16x2 arccos 4x
Ïðåäîñòåðåæåíèå
Èíîãäà áûâàåò, ÷òî ôóíêöèÿ îïðåäåëåíà â íåêîòîðîé òî÷êå, à å¼
ôîðìàëüíî âû÷èñëåííàÿ ïðîèçâîäíàÿ íåò.  òàêîé ñèòóàöèè
ïîëüçóþòñÿ íåïîñðåäñòâåííî îïðåäåëåíèåì ïðîèçâîäíîé.
Ïðèìåð 9. Ïðåäïîëîæèì ÷òî, íàì íóæíî âû÷èñëèòü
ïðîèçâîäíóþ ôóíêöèè
f (x) = x
√ √
p
(1 − x)2 sin x2
ïðè x ∈ (− 3, 3). Ôóíêöèÿ f íà ýòîì ïðîìåæóòêå êîððåêòíî
îïðåäåëåíà.
Åñëè ìû ôîðìàëüíî ïðèìåíèì ïðàâèëà äèôôåðåíöèðîâàíèÿ, òî
ïîëó÷èì
p
′
p
′
p
f ′ (x) = x (1 − x)2 sin x2 = x′ (1 − x)2 sin x2 +x
(1 − x)2 sin x2 =
′
p
x (1 − x)2 sin x2
2
2
= (1 − x) sin x + p
=
2 (1 − x)2 sin x2
p
x −2(1 − x) sin x2 + 2x(1 − x)2 cos x2
2
2
p
= (1 − x) sin x +
=
2 (1 − x)2 sin x2
p
x x(1 − x)2 cos x2 − (1 − x) sin x2
2
2
p
= (1 − x) sin x +
.
(1 − x)2 sin x2
Êàê ìû âèäèì,
ôóíêöèÿ íå îïðåäåëåíà â äâóõ òî÷êàõ
√
√ ïîëó÷åííàÿ
èíòåðâàëà (− 3, 3): ïðè x = 0 è ïðè x = 1.
f ′ (x0 ) = lim
x→x0
f (x) − f (x0 )
.
x − x0
Äëÿ x0 = 0 â ñèëó òîãî, ÷òî f (0) = 0, èìååì
lim
x→x0
p
f (x)
f (x) − f (x0 )
= lim
= lim (1 − x)2 sin x2 = 0.
x→0 x
x→0
x − x0
Ýòî çíà÷èò, ÷òî ôóíêöèÿ f äèôôåðåíöèðóåìà â íóëå è f ′ (0) = 0.
Äëÿ x0 = 1 â ñèëó òîãî, ÷òî f (1) = 0, èìååì
p
x (1 − x)2 sin x2
f (x) − f (x0 )
f (x)
lim
= lim
= lim
=
x→x0
x→1
x→1 x − 1
x − x0
x−1
√
√
x|x − 1| sin x2
= lim
= lim x sin x2 · sign (x − 1).
x→1
x→1
x−1
Êàê ìû âèäèì, ïîñëåäíèé ïðåäåë íå ñóùåñòâóåò, ïîñêîëüêó
lim sign (x−1) = −1,
x→1−
lim sign (x−1) = 1
x→1+
=⇒
∄ lim sign (x−1).
x→1
Ñëåäîâàòåëüíî, ôóíêöèÿ f íå äèôôåðåíöèðóåìà â òî÷êå 1.
Ïðîèçâîäíûå âûñøèõ ïîðÿäêîâ
Ïðîèçâîäíûå âûñøèõ ïîðÿäêîâ îïðåäåëÿþòñÿ èíäóêòèâíî: âòîðàÿ
ïðîèçâîäíàÿ ýòî ïåðâàÿ ïðîèçâîäíàÿ îò ïåðâîé ïðîèçâîäíîé è
òàê äàëåå:
′
f ′′ (x) = f ′ (x) ,
...,
′
f (n) (x) = f (n−1) (x) .
Òàêèì îáðàçîì, âû÷èñëåíèå n-îé ïðîèçâîäíîé ôóíêöèè f ýòî
ïðîñòî n-êðàòíîå âû÷èñëåíèå ïåðâîé ïðîèçâîäíîé.
Ïðèìåð 10. Äëÿ ôóíêöèè f (x) = sin x íàéä¼ì f (n).
π
f ′ (x) = cos x = sin x +
2
π
π
f ′′ (x) = cos x +
= sin x + 2 ·
2
2
π
π
′′
f (x) = cos x + 2 ·
= sin x + 3 ·
2
2
...
π(n − 1)
πn f (n) (x) = cos x +
= sin x +
2
2
Ïðîèçâîäíûå íåÿâíûõ ôóíêöèé
Ïóñòü íàì äàíî íåêîòîðîå óðàâíåíèå F (x, y) = 0.
Ïðåäïîëîæèì, ÷òî ìíîæåñòâî òî÷åê ïëîñêîñòè, êîîðäèíàòû
êîòîðûõ óäîâëåòâîðÿþò äàííîìó óðàâíåíèþ, ñîñòîèò èç êîíå÷íîãî
÷èñëà íåïðåðûâíûõ êðèâûõ, êàæäàÿ èç êîòîðûõ åñòü ãðàôèê
âçàèìíî îäíîçíà÷íîé ôóíêöèè y(x).
 òàêîì ñëó÷àå ãîâîðÿò, ÷òî óðàâíåíèå F (x, y) = 0
îïðåäåëÿåò ñîîòâåòñòâóþùåå ñåìåéñòâî ôóíêöèé
y1 (x), y2 (x), . . . , yn (x).
Åñëè òî÷êà (x0 , y0 ) ëåæèò òîëüêî íà îäíîé èç ýòèõ êðèâûõ, òî
óñëîâèå y(x0 ) = y0 ïîçâîëÿåò îäíîçíà÷íî âûáðàòü ýòó êðèâóþ èç
âñåãî ñåìåéñòâà. Èíûìè ñëîâàìè, óðàâíåíèå F (x, y) = 0 è óñëîâèå
y(x0 ) = y0 çàäàþò îäíîçíà÷íóþ
íåïðåðûâíóþ ôóíêöèþ â
îêðåñòíîñòè òî÷êè (x0 , y0 ) òàêóþ, ÷òî F (x, y(x)) = 0, y(x0 ) = y0 .
Èíà÷å ãîâîðÿ, åñëè óðàâíåíèå F (x, y) = 0 çàäà¼ò ¾äîñòàòî÷íî
õîðîøóþ¿ êðèâóþ íà ïëîñêîñòè, òî
(òî åñòü â
äîñòàòî÷íî ìàëîé îêðåñòíîñòè òî÷êè ýòîé êðèâîé) îíà ÿâëÿåòñÿ
ãðàôèêîì íåêîòîðîé ôóíêöèè y(x). À ýòî çíà÷èò, ÷òî å¼ ìîæíî
äèôôåðåíöèðîâàòü ïî ïåðåìåííîé x.
íåÿâíî
íåÿâíóþ
ëîêàëüíî
Ïðèìåð 11. Íàéä¼ì y è y , åñëè ôóíêöèÿ y(x) çàäàíà íåÿâíî
óðàâíåíèåì
′
′′
x4 + x2 y 2 + y = 4.
(1)
4x3 + 2xy 2 + 2x2 yy ′ + y ′ = 0.
(2)
Äëÿ ýòîãî ïðîäèôôåðåíöèðóåì îáå ÷àñòè ðàâåíñòâà (1), íå
çàáûâàÿ, òî y ýòî ôóíêöèÿ, çàâèñÿùàÿ îò x:
Èç ïîëó÷åííîãî ðàâåíñòâà âûðàæàåì y′ :
y′ =
−4x3 − 2xy 2
.
2x2 y + 1
Äëÿ íàõîæäåíèÿ y′′ ïðîäèôôåðåíöèðóåì îáå ÷àñòè ðàâåíñòâà (2)
è ïîäñòàâèì òóäà íàéäåííîå âûðàæåíèå äëÿ y′ :
12x2 + 2y 2 + 2x · 2yy ′ + 4xyy ′ + 2x2 (y ′ )2 + 2x2 yy ′′ + y ′′ = 0
⇐⇒
12x2 + 2y 2 + 8xyy ′ + 2x2 (y ′ )2 + 2x2 yy ′′ + y ′′ = 0 ⇐⇒
2
−4x3 − 2xy 2
−4x3 − 2xy 2
2
2
2
+2x
12x +2y +8xy
+2x2 yy ′′ +y ′′ = 0.
2x2 y + 1
2x2 y + 1
Îñòàëîñü ëèøü êàê è ðàíåå âûðàçèòü îòñþäà y′′ :
y ′′ =
−12x2 − 2y 2 +
8xy(4x3 +2xy 2 )
2x2 y+1
− 2x2
2x2 y + 1
−4x3 −2xy 2
2x2 y+1
2
.
Ïðèìåð 12. Íàéä¼ì y è y , åñëè ôóíêöèÿ y(x) çàäàíà íåÿâíî
óðàâíåíèåì
′
′′
tg y = 3x + 5y.
Äèôôåðåíöèðóåì âûðàæåíèå (3):
(3)
y′
= 3 + 5y ′ .
cos2 y
Âûðàæàÿ îòñþäà y′ , ïîëó÷àåì
y′ =
3
1
cos2 y
y′ =
−5
⇐⇒
3 cos2 y
.
1 − 5 cos2 y
(4)
×òîáû âû÷èñëèòü y′′ , ïðîäèôôåðåíöèðóåì ðàâåíñòâî (4):
′′
y =
3 cos2 y
1 − 5 cos2 y
=
′
=
−6 cos y sin yy ′ (1 − 5 cos2 y) − 30 cos3 y sin yy ′
=
(1 − 5 cos2 y)2
−6 cos y sin yy ′
(1 − 5 cos2 y)2
⇐⇒
y ′′ =
−18 cos3 y sin y
.
(1 − 5 cos2 y)3
Ïðîèçâîäíûå ôóíêöèé, çàäàííûõ ïàðàìåòðè÷åñêè
, íàçûâàåòñÿ ìíîæåñòâî òî÷åê
ïëîñêîñòè, êîîðäèíàòû (x, y) êîòîðûõ îïðåäåëÿþòñÿ èç
ñîîòíîøåíèé
(
Êðèâîé, çàäàííîé ïàðàìåòðè÷åñêè
x = x(t),
y = y(t)
ïðè êàæäîì ôèêñèðîâàííîì t èç íåêîòîðîãî ìíîæåñòâà
.
Ïóñòü ôóíêöèè x(t) è y(t) íåïðåðûâíû íà T , è ïóñòü ìíîæåñòâî T
ìîæíî ðàçáèòü íà êîíå÷íîå ÷èñëî ïðîìåæóòêîâ, íà êàæäîì èç
êîòîðûõ ôóíêöèÿ x(t) ñòðîãî ìîíîòîííà (è ïîòîìó âçàèìíî
îäíîçíà÷íà). Òîãäà íà êàæäîì òàêîì ïðîìåæóòêå îïðåäåëåíà
îáðàòíàÿ ôóíêöèÿ t(x) è, ñëåäîâàòåëüíî, ôóíêöèÿ y(t) = y(t(x))
íà í¼ì ÿâëÿåòñÿ ôóíêöèåé ïåðåìåííîé x. Èíûìè ñëîâàìè,
(íà êàæäîì ïðîìåæóòêå ìîíîòîííîñòè x(t)) y ìîæíî
ðàññìàòðèâàòü êàê ôóíêöèþ îò x.
T ⊂ (−∞, +∞)
ëîêàëüíî
Èç ðàâåíñòâà y(t) = y(t(x)) ïî ïðàâèëó äèôôåðåíöèðîâàíèÿ
ñëîæíîé ôóíêöèè ïîëó÷àåì
yx′ = yt′ · t′x .
Ïî ïðàâèëó äèôôåðåíöèðîâàíèÿ îáðàòíîé ôóíêöèè t′x = x1′ , à
t
òîãäà
′
yx′ =
yt
.
x′t
(5)
Äëÿ âû÷èñëåíèÿ yx′′ ïðîñòî ïðîäèôôåðåíöèðóåì ðàâåíñòâî (5):
yx′′ =
yt′
x′t
′
=
x
yt′
x′t
′
t
yx′′ =
· t′x =
yt′′ x′t − yt′ x′′t 1
· ′
(x′t )2
xt
yt′′ x′t − yt′ x′′t
.
(x′t )3
Îáû÷íî èíäåêñ x â îáîçíà÷åíèÿõ yx′ è yx′′ îïóñêàþò.
⇐⇒
(6)
Ïðèìåð 13. Íàéä¼ì y è y äëÿ ôóíêöèè y(x), çàäàííîé
′
ïàðàìåòðè÷åñêè:
′′
(
x(t) = e−2t ,
y(t) = e4t .
x′t = −2e−2t ,
x′′t = 4e−2t ,
yt′ = 4e4t ,
yt′′ = 16e4t .
Ïî ôîðìóëàì (5) è (7) ïîëó÷àåì
y′ =
y ′′ =
4e4t
yt′
=
= −2e6t ,
x′t
−2e−2t
16e4t · (−2e−2t ) − 4e4t · 4e−2t
yt′′ x′t − yt′ x′′t
=
=
′
3
(xt )
(−2e−2t )3
=
−32e2t − 16e2t
= 6e8t .
−8e−6t
Ãåîìåòðè÷åñêèé ñìûñë ïðîèçâîäíîé
Âñå çíàþò, ÷òî
y = f (x)
x0
f ′ (x0 ).
Îòñþäà, â ÷àñòíîñòè, ñëåäóþò èçâåñòíûå âûðàæåíèÿ äëÿ
êàñàòåëüíîé è íîðìàëè ê êðèâîé y = f (x) â òî÷êå x0 :
y − y0 = f ′ (x0 )(x − x0 ) óðàâíåíèå êàñàòåëüíîé;
1
(x − x0 ) óðàâíåíèå íîðìàëè
y − y0 = − ′
f (x )
òàíãåíñ óãëà íàêëîíà êàñàòåëüíîé ê êðèâîé
â òî÷êå
ðàâåí ïðîèçâîäíîé
0
(çäåñü y0 = f (x0 )).
Ïðè f ′ (x0 ) = 0 óðàâíåíèå íîðìàëè èìååò âèä x = x0 .
Ïðèìåð 14. Íàïèøåì óðàâíåíèÿ êàñàòåëüíûõ è íîðìàëåé ê
êðèâîé y = 4x − x3 â òî÷êàõ å¼ ïåðåñå÷åíèÿ ñ îñüþ Ox.
Òàêèõ òî÷åê, î÷åâèäíî, òðè: x0 = 0 è x1,2 = ±2.
Ïîñêîëüêó äëÿ f (x) = 4x − x3 èìååì f ′ (x) = 4 − 3x2 , íàõîäèì
f ′ (x0 ) = 4,
f ′ (x1,2 ) = −8.
Òàê êàê y0 = y1 = y2 = 0 ïî óñëîâèþ (ýòî îðäèíàòû òî÷åê, â
êîòîðûõ ôóíêöèÿ ïåðåñåêàåò îñü àáñöèññ), îòñþäà ïîëó÷àåì, ÷òî
èñêîìûå êàñàòåëüíûå è íîðìàëè èìåþò âèä
y = 4x óðàâíåíèå êàñàòåëüíîé â òî÷êå x0 ,
x
óðàâíåíèå íîðìàëè â òî÷êå x0 ;
y=−
4
y = −8(x ∓ 2) óðàâíåíèå êàñàòåëüíûõ â òî÷êàõ x1 è x2 ,
1
y = (x ∓ 2) óðàâíåíèå íîðìàëåé â òî÷êàõ x1 è x2 .
8
Ïðèìåð 15. Îïðåäåëèì óãëîâîé êîýôôèöèåíò êàñàòåëüíîé ê
êðèâîé
x2 − y 2 + xy − 11 = 0
â òî÷êå (3, 2).
Êàê ìû âèäèì, êðèâàÿ çàäàíà íåÿâíî, îäíàêî ãåîìåòðè÷åñêèé
ñìûñë ïðîèçâîäíîé íåÿâíîé ôóíêöèè òî÷íî òàêîé æå, êàê è ó
ôóíêöèè, çàäàííîé ÿâíî.
Äèôôåðåíöèðóÿ ðàâåíñòâî, êîòîðûì çàäàíà êðèâàÿ, ïîëó÷àåì
2x − 2yy ′ + y + xy ′ = 0,
òî åñòü
y′ =
2x + y
.
2y − x
Ïîäñòàâëÿÿ ñþäà x = 3, y = 2, ïîëó÷àåì y′ = 8. Ýòî çíà÷èò, ÷òî
óãëîâîé êîýôôèöèåíò êàñàòåëüíîé ê ðàññìàòðèâàåìîé êðèâîé â
òî÷êå (3, 2) ðàâåí arctg 8.
Äèôôåðåíöèàë è ïðèáëèæ¼ííûå âû÷èñëåíèÿ
Íàïîìíþ, ÷òî ôóíêöèÿ f íàçûâàåòñÿ äèôôåðåíöèðóåìîé â òî÷êå
, åñëè âûïîëíÿåòñÿ óñëîâèå
x0
f (x) = f (x0 ) + f ′ (x0 )(x − x0 ) + o(x − x0 ),
x → x0 .
Îáîçíà÷èì ∆x = x − x0 , ∆f (x) = f (x) − f (x0 ), òîãäà ýòî
ðàâåíñòâî ìîæíî ïåðåïèñàòü â âèäå
∆f (x) = f ′ (x0 )∆x + o (∆x) ,
∆x → 0.
ôóíêöèè f â òî÷êå x0 íàçûâàåòñÿ ãëàâíàÿ ÷àñòü
å¼ ïðèðàùåíèÿ ∆f (x), ëèíåéíî çàâèñÿùàÿ îò ïðèðàùåíèÿ
àðãóìåíòà ∆x. Èíûìè ñëîâàìè, äëÿ äèôôåðåíöèàëà ôóíêöèè f â
òî÷êå x0 , îáîçíà÷àåìîãî df (x0 ), ñïðàâåäëèâà ôîðìóëà
Äèôôåðåíöèàëîì
df (x0 ) = f ′ (x0 )∆x.
Äëÿ òîæäåñòâåííîé ôóíêöèè f (x) = x äèôôåðåíöèàë â ëþáîé
òî÷êå ñîâïàäàåò ñ ïðèðàùåíèåì: df (x) = dx = ∆x, ïîýòîìó îáû÷íî
âìåñòî îáîçíà÷åíèÿ ∆x ïèøóò dx.
Ïîñêîëüêó äèôôåðåíöèàë ôóíêöèè îòëè÷àåòñÿ îò å¼ ïðèðàùåíèÿ
íà áåñêîíå÷íî ìàëóþ áîëåå âûñîêîãî ïîðÿäêà, ÷åì dx, äëÿ
äîñòàòî÷íî ìàëûõ dx ñïðàâåäëèâî ïðèáëèæ¼ííîå ðàâåíñòâî
∆f ≈ df , êîòîðîå ðàâíîñèëüíî
f (x) ≈ f (x0 ) + f ′ (x0 )(x − x0 ).
(7)
Ôîðìóëà (7) ïîçâîëÿåò ïðèáëèæ¼ííî âû÷èñëÿòü çíà÷åíèÿ
ôóíêöèè â òî÷êå x, åñëè ìû çíàåì çíà÷åíèå ýòîé ôóíêöèè è å¼
ïðîèçâîäíîé â äîñòàòî÷íî áëèçêîé ê x òî÷êå x0 .
Ïðèìåð 16. Âû÷èñëèì ïðèáëèæ¼ííî (3, 03) .
5
Äëÿ ýòîãî ðàññìîòðèì ôóíêöèþ f (x) = x è âîñïîëüçóåìñÿ
ôîðìóëîé (7) ïðè x = 3, 03 è x0 = 3. Èìååì
5
f (x0 ) = 243,
f ′ (x) = 5x4 ,
f ′ (x0 ) = 405,
x − x0 = 0, 03,
ïîýòîìó
(3, 03)5 = 243 + 405 · 0, 03 = 243 + 12, 15 = 255, 15.