Маслова Т.Н., Суходский А.М. Справочник школьника по математике. 5-11 кл
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ÀËÃÅÁÐÀ
§ 23. Ïðîèçâîäíàÿ
ÀËÃÅÁÐÀ
305304
Ðàçäåë VII. ÝËÅÌÅÍÒÛ ÌÀÒ. ÀÍÀËÈÇÀ
Èòàê,
=---=
--
+
-®-®
)
37
(
lim
37
)
2
(
lim
22
x
x
x
xx
.6)327( -=++-=
n
§ 23. Ïðîèçâîäíàÿ
220. Ïðèðàùåíèå àðãóìåíòà. Ïðèðàùåíèå ôóí-
êöèè. Ïóñòü ôóíêöèÿ
)(xfy =
îïðåäåëåíà â òî÷êàõ
õ è õ
1
. Ðàçíîñòü õ
1
õ íàçûâàåòñÿ ïðèðàùåíèåì
àðãóìåíòà, à ðàçíîñòü
)()(
1
xfxf -
ïðèðàùåíè-
åì ôóíêöèè ïðè ïåðåõîäå îò çíà÷åíèÿ àðãóìåíòà õ
ê çíà÷åíèþ àðãóìåíòà õ
1
.
Ïðèðàùåíèå àðãóìåíòà îáîçíà÷àþò
;xD
çíà÷èò,
,
1
xxx -=D
ò. å.
.
1
xxx D+=
Ïðèðàùåíèå ôóíêöèè îáîçíà÷àþò
fD
èëè
;yD
çíà÷èò,
).()()()(
1
xfxxfxfxff -D+=-=D
Ï ð è ì å ð 1. Íàéòè ïðèðàùåíèå ôóíêöèè ó =
= õ
3
ïðè ïåðåõîäå îò çíà÷åíèÿ àðãóìåíòà õ ê çíà÷å-
íèþ
.xx D+
q Èìååì
.)()(,)(
33
xxxxfxxf D+=D+=
Ïî-
ýòîìó
Df = f(x + Dx) f(x) = (x + Dx)
3
x
3
= x
3
+ 3x
3
· Dx +
.)()(33)()(3
322332
xxxxxxxxx D+D+D=-D+D+
(ñì. ï. 216); ïðÿìàÿ õ = 3 ÿâëÿåòñÿ âåðòèêàëüíîé
àñèìïòîòîé ãðàôèêà ôóíêöèè
.
65
9
2
2
+-
+
=
xx
x
y
â) Çäåñü â îòëè÷èå îò ïðåäûäóùåãî ïðèìåðà è
÷èñëèòåëü, è çíàìåíàòåëü îáðàùàþòñÿ â íóëü ïðè
õ = 3.  ïîäîáíûõ ñëó÷àÿõ äëÿ âû÷èñëåíèÿ ïðåäåëà
íåîáõîäèìû òîæäåñòâåííûå ïðåîáðàçîâàíèÿ âûðàæå-
íèÿ, çàäàþùåãî ôóíêöèþ. Èìååì
=
+-
-
65
9
2
2
xx
x
.
)
2
(
)
3
(
)
3
(
)
3
(
--
+-
=
xx
xx
Ïîñêîëüêó ïðè
3®x
çíà÷åíèå
ôóíêöèè â ñàìîé òî÷êå õ = 3 âî âíèìàíèå íå ïðèíè-
ìàåòñÿ (ñì. ï. 217), äðîáü ìîæíî ñîêðàòèòü íà õ 3,
è òîãäà ïîëó÷èì
.
2
3
-
+
x
x
Èòàê,
=
--
+-
=
+-
-
®®
)
2
(
)
3
(
)
3
(
)
3
(
lim
65
9
lim
3
2
2
3
xx
xx
xx
x
xx
.6
23
33
2
3
lim
3
=
-
+
=
-
+
=
®
x
x
x
ã) Ïðè õ = 2 è ÷èñëèòåëü, è çíàìåíàòåëü îáðàùà-
þòñÿ â íóëü. Âûïîëíèì ñëåäóþùèå ïðåîáðàçîâàíèÿ
çàäàííîãî âûðàæåíèÿ:
=
+---
+-+
=
--
+
)37)(37(
)37)(2(
37
2
xx
xx
x
x
).37(
)2(
)37)(2(
3)7(
)37)(2(
22
+--=
+-
+-+
=
--
+-+
= x
x
xx
x
xx
ÀËÃÅÁÐÀ
§ 23. Ïðîèçâîäíàÿ
ÀËÃÅÁÐÀ
305304
Ðàçäåë VII. ÝËÅÌÅÍÒÛ ÌÀÒ. ÀÍÀËÈÇÀ
Èòàê,
=---=
--
+
-®-®
)
37
(
lim
37
)
2
(
lim
22
x
x
x
xx
.6)327( -=++-=
n
§ 23. Ïðîèçâîäíàÿ
220. Ïðèðàùåíèå àðãóìåíòà. Ïðèðàùåíèå ôóí-
êöèè. Ïóñòü ôóíêöèÿ
)(xfy =
îïðåäåëåíà â òî÷êàõ
õ è õ
1
. Ðàçíîñòü õ
1
õ íàçûâàåòñÿ ïðèðàùåíèåì
àðãóìåíòà, à ðàçíîñòü
)()(
1
xfxf -
ïðèðàùåíè-
åì ôóíêöèè ïðè ïåðåõîäå îò çíà÷åíèÿ àðãóìåíòà õ
ê çíà÷åíèþ àðãóìåíòà õ
1
.
Ïðèðàùåíèå àðãóìåíòà îáîçíà÷àþò
;xD
çíà÷èò,
,
1
xxx -=D
ò. å.
.
1
xxx D+=
Ïðèðàùåíèå ôóíêöèè îáîçíà÷àþò
fD
èëè
;yD
çíà÷èò,
).()()()(
1
xfxxfxfxff -D+=-=D
Ï ð è ì å ð 1. Íàéòè ïðèðàùåíèå ôóíêöèè ó =
= õ
3
ïðè ïåðåõîäå îò çíà÷åíèÿ àðãóìåíòà õ ê çíà÷å-
íèþ
.xx D+
q Èìååì
.)()(,)(
33
xxxxfxxf D+=D+=
Ïî-
ýòîìó
Df = f(x + Dx) f(x) = (x + Dx)
3
x
3
= x
3
+ 3x
3
· Dx +
.)()(33)()(3
322332
xxxxxxxxx D+D+D=-D+D+
(ñì. ï. 216); ïðÿìàÿ õ = 3 ÿâëÿåòñÿ âåðòèêàëüíîé
àñèìïòîòîé ãðàôèêà ôóíêöèè
.
65
9
2
2
+-
+
=
xx
x
y
â) Çäåñü â îòëè÷èå îò ïðåäûäóùåãî ïðèìåðà è
÷èñëèòåëü, è çíàìåíàòåëü îáðàùàþòñÿ â íóëü ïðè
õ = 3.  ïîäîáíûõ ñëó÷àÿõ äëÿ âû÷èñëåíèÿ ïðåäåëà
íåîáõîäèìû òîæäåñòâåííûå ïðåîáðàçîâàíèÿ âûðàæå-
íèÿ, çàäàþùåãî ôóíêöèþ. Èìååì
=
+-
-
65
9
2
2
xx
x
.
)
2
(
)
3
(
)
3
(
)
3
(
--
+-
=
xx
xx
Ïîñêîëüêó ïðè
3®x
çíà÷åíèå
ôóíêöèè â ñàìîé òî÷êå õ = 3 âî âíèìàíèå íå ïðèíè-
ìàåòñÿ (ñì. ï. 217), äðîáü ìîæíî ñîêðàòèòü íà õ 3,
è òîãäà ïîëó÷èì
.
2
3
-
+
x
x
Èòàê,
=
--
+-
=
+-
-
®®
)
2
(
)
3
(
)
3
(
)
3
(
lim
65
9
lim
3
2
2
3
xx
xx
xx
x
xx
.6
23
33
2
3
lim
3
=
-
+
=
-
+
=
®
x
x
x
ã) Ïðè õ = 2 è ÷èñëèòåëü, è çíàìåíàòåëü îáðàùà-
þòñÿ â íóëü. Âûïîëíèì ñëåäóþùèå ïðåîáðàçîâàíèÿ
çàäàííîãî âûðàæåíèÿ:
=
+---
+-+
=
--
+
)37)(37(
)37)(2(
37
2
xx
xx
x
x
).37(
)2(
)37)(2(
3)7(
)37)(2(
22
+--=
+-
+-+
=
--
+-+
= x
x
xx
x
xx
ÀËÃÅÁÐÀ
§ 23. Ïðîèçâîäíàÿ
ÀËÃÅÁÐÀ
307306
Ðàçäåë VII. ÝËÅÌÅÍÒÛ ÌÀÒ. ÀÍÀËÈÇÀ
ïðåäåë íàçûâàåòñÿ çíà÷åíèåì ïðîèçâîäíîé ôóíê-
öèè
)(xfy =
â òî÷êå õ è îáîçíà÷àåòñÿ
)(xf
¢
èëè
.y
¢
Èòàê,
.
)
(
)
(
limlim
)
(
00
x
xfxxf
x
f
yxf
xx
D
-D+
=
D
D
=
¢
=
¢
®D®D
Îòìåòèì, ÷òî
)(xf
¢
ýòî íîâàÿ ôóíêöèÿ, îïðåäå-
ëåííàÿ âî âñåõ òàêèõ òî÷êàõ õ, â êîòîðûõ ñóùåñòâó-
åò óêàçàííûé âûøå ïðåäåë; åå íàçûâàþò ïðîèçâîä-
íîé ôóíêöèè
)(xfy =
.
Ï ð è ì å ð 1. Íàéòè
),2(f
¢
åñëè
.)(
2
xxf =
q Èìååì
,)2()2(,42)2(
22
xxff D+=D+==
=Df
.)(44)2()2()2(
22
xxxfxf D+D=-D+=-D+=
Òîãäà
,4
)(4
2
x
x
xx
x
f
D+=
D
D+D
=
D
D
.4
)
4
(
limlim
00
=D+=
D
D
®D®D
x
x
f
xx
Çíà÷èò,
.4)2( =
¢
f
n
Èñïîëüçóÿ îïðåäåëåíèå, ìîæíî ðåêîìåíäîâàòü ñëå-
äóþùåå
ïðàâèëî îòûñêàíèÿ ïðîèçâîäíîé ôóíêöèè
y = f (x):
1. Ôèêñèðóþò çíà÷åíèå õ è íàõîäÿò
).(xf
2. Äàþò àðãóìåíòó õ ïðèðàùåíèå
xD
è íàõîäÿò
).( xxf D+
3. Âû÷èñëÿþò ïðèðàùåíèå ôóíêöèè +=D xff (
).() xfx -D+
4. Ñîñòàâëÿþò îòíîøåíèå
.
x
f
D
D
Èòàê,
.))(33(
22
xxxxxf DD+D×+=D
n
Ïî ýòîé ôîðìóëå ìîæíî âû÷èñëÿòü çíà÷åíèå
fD
äëÿ ëþáûõ çàäàííûõ õ è
.xD
Íàïðèìåð, ïðè õ = 1,
2,0-=Dx
ïîëó÷àåì
=-=D )1()8,0( fff
.488,0)2,0())2,0()2,0(1313(
22
-=-×-+-××+×=
Ï ð è ì å ð 2. Äîêàçàòü, ÷òî äëÿ ëèíåéíîé ôóíê-
öèè
bkxy +=
ñïðàâåäëèâî ðàâåíñòâî
.
x
y
k
D
D
=
q Èìååì +=D++= xkxxfbkxxf ()(,)( Dx) + b.
Çíà÷èò,
,)())(()()( xkbkxbxxkxfxxfy D=+-+D+=-D+=D
îòêóäà ïîëó÷àåì
.k
x
y
=
D
D
n
221. Îïðåäåëåíèå ïðîèçâîäíîé. Ïóñòü ôóíêöèÿ
)(xfy =
îïðåäåëåíà â òî÷êå õ è â íåêîòîðîé îêðåñò-
íîñòè ýòîé òî÷êè. Ïóñòü, äàëåå,
xD
ïðèðàùåíèå
àðãóìåíòà, ïðè÷åì òàêîå, ÷òî òî÷êà
xx D+
ïðèíàä-
ëåæèò óêàçàííîé îêðåñòíîñòè òî÷êè õ, à
fD
ñîîò-
âåòñòâóþùåå ïðèðàùåíèå ôóíêöèè, ò. å.
=Df
).()( xfxxf -D+=
Åñëè ñóùåñòâóåò ïðåäåë îòíîøå-
íèÿ ïðèðàùåíèÿ ôóíêöèè
fD
ê ïðèðàùåíèþ àðãó-
ìåíòà
xD
ïðè óñëîâèè
,0®Dx
òî ôóíêöèÿ
)(xfy =
íàçûâàåòñÿ äèôôåðåíöèðóåìîé â òî÷êå õ, à ýòîò
ÀËÃÅÁÐÀ
§ 23. Ïðîèçâîäíàÿ
ÀËÃÅÁÐÀ
307306
Ðàçäåë VII. ÝËÅÌÅÍÒÛ ÌÀÒ. ÀÍÀËÈÇÀ
ïðåäåë íàçûâàåòñÿ çíà÷åíèåì ïðîèçâîäíîé ôóíê-
öèè
)(xfy =
â òî÷êå õ è îáîçíà÷àåòñÿ
)(xf
¢
èëè
.y
¢
Èòàê,
.
)
(
)
(
limlim
)
(
00
x
xfxxf
x
f
yxf
xx
D
-D+
=
D
D
=
¢
=
¢
®D®D
Îòìåòèì, ÷òî
)(xf
¢
ýòî íîâàÿ ôóíêöèÿ, îïðåäå-
ëåííàÿ âî âñåõ òàêèõ òî÷êàõ õ, â êîòîðûõ ñóùåñòâó-
åò óêàçàííûé âûøå ïðåäåë; åå íàçûâàþò ïðîèçâîä-
íîé ôóíêöèè
)(xfy =
.
Ï ð è ì å ð 1. Íàéòè
),2(f
¢
åñëè
.)(
2
xxf =
q Èìååì
,)2()2(,42)2(
22
xxff D+=D+==
=Df
.)(44)2()2()2(
22
xxxfxf D+D=-D+=-D+=
Òîãäà
,4
)(4
2
x
x
xx
x
f
D+=
D
D+D
=
D
D
.4
)
4
(
limlim
00
=D+=
D
D
®D®D
x
x
f
xx
Çíà÷èò,
.4)2( =
¢
f
n
Èñïîëüçóÿ îïðåäåëåíèå, ìîæíî ðåêîìåíäîâàòü ñëå-
äóþùåå
ïðàâèëî îòûñêàíèÿ ïðîèçâîäíîé ôóíêöèè
y = f (x):
1. Ôèêñèðóþò çíà÷åíèå õ è íàõîäÿò
).(xf
2. Äàþò àðãóìåíòó õ ïðèðàùåíèå
xD
è íàõîäÿò
).( xxf D+
3. Âû÷èñëÿþò ïðèðàùåíèå ôóíêöèè +=D xff (
).() xfx -D+
4. Ñîñòàâëÿþò îòíîøåíèå
.
x
f
D
D
Èòàê,
.))(33(
22
xxxxxf DD+D×+=D
n
Ïî ýòîé ôîðìóëå ìîæíî âû÷èñëÿòü çíà÷åíèå
fD
äëÿ ëþáûõ çàäàííûõ õ è
.xD
Íàïðèìåð, ïðè õ = 1,
2,0-=Dx
ïîëó÷àåì
=-=D )1()8,0( fff
.488,0)2,0())2,0()2,0(1313(
22
-=-×-+-××+×=
Ï ð è ì å ð 2. Äîêàçàòü, ÷òî äëÿ ëèíåéíîé ôóíê-
öèè
bkxy +=
ñïðàâåäëèâî ðàâåíñòâî
.
x
y
k
D
D
=
q Èìååì +=D++= xkxxfbkxxf ()(,)( Dx) + b.
Çíà÷èò,
,)())(()()( xkbkxbxxkxfxxfy D=+-+D+=-D+=D
îòêóäà ïîëó÷àåì
.k
x
y
=
D
D
n
221. Îïðåäåëåíèå ïðîèçâîäíîé. Ïóñòü ôóíêöèÿ
)(xfy =
îïðåäåëåíà â òî÷êå õ è â íåêîòîðîé îêðåñò-
íîñòè ýòîé òî÷êè. Ïóñòü, äàëåå,
xD
ïðèðàùåíèå
àðãóìåíòà, ïðè÷åì òàêîå, ÷òî òî÷êà
xx D+
ïðèíàä-
ëåæèò óêàçàííîé îêðåñòíîñòè òî÷êè õ, à
fD
ñîîò-
âåòñòâóþùåå ïðèðàùåíèå ôóíêöèè, ò. å.
=Df
).()( xfxxf -D+=
Åñëè ñóùåñòâóåò ïðåäåë îòíîøå-
íèÿ ïðèðàùåíèÿ ôóíêöèè
fD
ê ïðèðàùåíèþ àðãó-
ìåíòà
xD
ïðè óñëîâèè
,0®Dx
òî ôóíêöèÿ
)(xfy =
íàçûâàåòñÿ äèôôåðåíöèðóåìîé â òî÷êå õ, à ýòîò
ÀËÃÅÁÐÀ
§ 23. Ïðîèçâîäíàÿ
ÀËÃÅÁÐÀ
309308
Ðàçäåë VII. ÝËÅÌÅÍÒÛ ÌÀÒ. ÀÍÀËÈÇÀ
4.
.)(33
22
xxxx
x
f
D+D×+=
D
D
5.
=D+D×+=
D
D
®D®D
)
)
(
33
(
limlim
22
00
xxxx
x
f
xx
.30033
222
xxx =+×+=
Èòàê,
.3)(
23
xx =
¢
n
222. Ôîðìóëû äèôôåðåíöèðîâàíèÿ. Òàáëèöà
ïðîèçâîäíûõ. Îïåðàöèþ îòûñêàíèÿ ïðîèçâîäíîé íà-
çûâàþò äèôôåðåíöèðîâàíèåì.  ï. 221 ïîëó÷åíà
îäíà èç ôîðìóë äèôôåðåíöèðîâàíèÿ:
.3)(
23
xx =
¢
Ïî
òàêîìó æå ïëàíó ìîæíî âûâåñòè ôîðìóëû, êîòîðûå
ïðèâåäåíû â ñëåäóþùåé
òàáëèöå ïðîèçâîäíûõ:
5. Íàõîäÿò ïðåäåë îòíîøåíèÿ
x
f
D
D
ïðè
.0®Dx
Îòíîøåíèå
x
f
D
D
íàçûâàþò ðàçíîñòíûì îòíîøå-
íèåì. Îíî âûðàæàåò ñðåäíþþ ñêîðîñòü èçìåíåíèÿ
ôóíêöèè f íà ïðîìåæóòêå ñ êîíöàìè â òî÷êàõ õ è
.xx D+
Ìîæíî ñêàçàòü, ÷òî ïðîèçâîäíîé ôóíêöèè f
â òî÷êå õ íàçûâàåòñÿ ÷èñëî, ê êîòîðîìó ñòðåìèòñÿ
ðàçíîñòíîå îòíîøåíèå
x
f
D
D
ïðè
.0®Dx
Èíîãäà âìåñòî
xD
ïèøóò h, è òîãäà îïðåäåëåíèå
ïðîèçâîäíîé çàïèñûâàåòñÿ ñëåäóþùèì îáðàçîì:
.
)
(
)
(
lim
)
(
0
h
xfhxf
xf
h
-+
=
¢
®
Èíîãäà âìåñòî
xx D+
ïèøóò õ
1
, è òîãäà îïðåäå-
ëåíèå ïðîèçâîäíîé çàïèñûâàåòñÿ òàê:
.
)()(
lim
)(
1
1
1
xx
xfxf
xf
xx
-
-
=
¢
®
Ï ð è ì å ð 2. Íàéòè ïðîèçâîäíóþ ôóíêöèè õ
3
.
q Èñïîëüçóÿ ïðàâèëî îòûñêàíèÿ ïðîèçâîäíîé, ïîñ-
ëåäîâàòåëüíî íàõîäèì:
1.
.)(
3
xxf =
2.
.)()(
3
xxxxf D+=D+
3.
=-D+=-D+=D
33
)()()(
xxxxfxxff
.))(33(
22
xxxxx DD+D×+=
9.
.sin)(cos xx -=
¢
10.
.
cos
1
)
(
tg
2
x
x =
¢
11.
.
sin
1
)
(
ctg
2
x
x -=
¢
12.
.
1
1
)
(
arcsin
2
x
x
-
=
¢
13.
.
1
1
)
(
arccos
2
x
x
-
-=
¢
14.
.
1
1
)
(
arctg
2
x
x
+
=
¢
15.
.
1
1
)
(
arcctg
2
x
x
+
-=
¢
1.
.0=
¢
C
2.
.)( kbkx =
¢
+
3.
.)(
1
-
=
¢
rr
rxx
4.
.)(
xx
ee
=
¢
5.
.ln)( aaa
xx
=
¢
6.
.
1
)(ln
x
x =
¢
7.
.
ln
1
)
(
log
ax
x
a
=
¢
8.
.cos)(sin xx =
¢
ÀËÃÅÁÐÀ
§ 23. Ïðîèçâîäíàÿ
ÀËÃÅÁÐÀ
309308
Ðàçäåë VII. ÝËÅÌÅÍÒÛ ÌÀÒ. ÀÍÀËÈÇÀ
4.
.)(33
22
xxxx
x
f
D+D×+=
D
D
5.
=D+D×+=
D
D
®D®D
)
)
(
33
(
limlim
22
00
xxxx
x
f
xx
.30033
222
xxx =+×+=
Èòàê,
.3)(
23
xx =
¢
n
222. Ôîðìóëû äèôôåðåíöèðîâàíèÿ. Òàáëèöà
ïðîèçâîäíûõ. Îïåðàöèþ îòûñêàíèÿ ïðîèçâîäíîé íà-
çûâàþò äèôôåðåíöèðîâàíèåì.  ï. 221 ïîëó÷åíà
îäíà èç ôîðìóë äèôôåðåíöèðîâàíèÿ:
.3)(
23
xx =
¢
Ïî
òàêîìó æå ïëàíó ìîæíî âûâåñòè ôîðìóëû, êîòîðûå
ïðèâåäåíû â ñëåäóþùåé
òàáëèöå ïðîèçâîäíûõ:
5. Íàõîäÿò ïðåäåë îòíîøåíèÿ
x
f
D
D
ïðè
.0®Dx
Îòíîøåíèå
x
f
D
D
íàçûâàþò ðàçíîñòíûì îòíîøå-
íèåì. Îíî âûðàæàåò ñðåäíþþ ñêîðîñòü èçìåíåíèÿ
ôóíêöèè f íà ïðîìåæóòêå ñ êîíöàìè â òî÷êàõ õ è
.xx D+
Ìîæíî ñêàçàòü, ÷òî ïðîèçâîäíîé ôóíêöèè f
â òî÷êå õ íàçûâàåòñÿ ÷èñëî, ê êîòîðîìó ñòðåìèòñÿ
ðàçíîñòíîå îòíîøåíèå
x
f
D
D
ïðè
.0®Dx
Èíîãäà âìåñòî
xD
ïèøóò h, è òîãäà îïðåäåëåíèå
ïðîèçâîäíîé çàïèñûâàåòñÿ ñëåäóþùèì îáðàçîì:
.
)(
)
(
lim
)
(
0
h
xfhxf
xf
h
-+
=
¢
®
Èíîãäà âìåñòî
xx D+
ïèøóò õ
1
, è òîãäà îïðåäå-
ëåíèå ïðîèçâîäíîé çàïèñûâàåòñÿ òàê:
.
)()(
lim
)(
1
1
1
xx
xfxf
xf
xx
-
-
=
¢
®
Ï ð è ì å ð 2. Íàéòè ïðîèçâîäíóþ ôóíêöèè õ
3
.
q Èñïîëüçóÿ ïðàâèëî îòûñêàíèÿ ïðîèçâîäíîé, ïîñ-
ëåäîâàòåëüíî íàõîäèì:
1.
.)(
3
xxf =
2.
.)()(
3
xxxxf D+=D+
3.
=-D+=-D+=D
33
)()()(
xxxxfxxff
.))(33(
22
xxxxx DD+D×+=
9.
.sin)(cos xx -=
¢
10.
.
cos
1
)
(
tg
2
x
x =
¢
11.
.
sin
1
)
(
ctg
2
x
x -=
¢
12.
.
1
1
)
(
arcsin
2
x
x
-
=
¢
13.
.
1
1
)
(
arccos
2
x
x
-
-=
¢
14.
.
1
1
)
(
arctg
2
x
x
+
=
¢
15.
.
1
1
)
(
arcctg
2
x
x
+
-=
¢
1.
.0=
¢
C
2.
.)( kbkx =
¢
+
3.
.)(
1
-
=
¢
rr
rxx
4.
.)(
xx
ee
=
¢
5.
.ln)( aaa
xx
=
¢
6.
.
1
)(ln
x
x =
¢
7.
.
ln
1
)
(
log
ax
x
a
=
¢
8.
.cos)(sin xx =
¢
ÀËÃÅÁÐÀ
§ 23. Ïðîèçâîäíàÿ
ÀËÃÅÁÐÀ
311310
Ðàçäåë VII. ÝËÅÌÅÍÒÛ ÌÀÒ. ÀÍÀËÈÇÀ
Ò.7.8. Ïðîèçâîäíàÿ ïðîèçâåäåíèÿ äâóõ ôóíêöèé âû-
÷èñëÿåòñÿ ïî ôîðìóëå
vuvuuv
¢
+
¢
=
¢
)(
(òåîðåìà î äèôôåðåíöèðîâàíèè ïðîèçâåäåíèÿ).
Ò.7.9. Ïðîèçâîäíàÿ ÷àñòíîãî äâóõ ôóíêöèé âû÷èñ-
ëÿåòñÿ ïî ôîðìóëå
2
v
vuvu
v
u
¢
-
¢
=
¢
÷
ø
ö
ç
è
æ
ïðè óñëîâèè
0)( ¹xv
(òåîðåìà î äèôôåðåíöèðîâàíèè ÷àñòíîãî).
Ï ð è ì å ð 1. Íàéòè ïðîèçâîäíóþ ôóíêöèè:
à)
;5cos7,0sin2 +-= xxy
á)
.log
3
4,0
xx
q à) Èñïîëüçóÿ òåîðåìû 7.6 è 7.7, èìååì
=
¢
+
¢
-+
¢
=
¢
+- 5)cos7,0()sin2()5cos7,0sin2( xxxx
.5)(cos7,0)(sin2
¢
+
¢
-
¢
= xx
Îñòàåòñÿ ïðèìåíèòü ñîîòâåòñòâóþùèå ôîðìóëû
äèôôåðåíöèðîâàíèÿ (ñì. ï. 222). Òîãäà ïîëó÷èì
.sin7,0cos20)sin(7,0cos2 xxxx +=+--
á) Ñîãëàñíî òåîðåìå 7.8, íàõîäèì
.
)
(
loglog
)
(
)
log
(
3
4,0
3
4,0
3
4,0
¢
+
¢
=
¢
xxxxxx
Òåïåðü ïðèìåíèì ôîðìóëû äèôôåðåíöèðîâàíèÿ
è ïîëó÷èì
.
3ln
1ln4,0
3ln
1
log4,0
4,0
4,0
3
6,0
×
+
=×+
-
x
x
x
xxx
n
Íàïðèìåð:
2)32( =
¢
-x
(ïî ôîðìóëå 2);
,2
)
(
1
,10
)
(
32
2
910
--
-=
¢
=
¢
÷
÷
ø
ö
ç
ç
è
æ
=
¢
xx
x
xx
5
2
5
3
5
3
)
(
-
=
¢
xx
(ïî ôîðìóëå 3);
6ln6)6(
xx
=
¢
(ïî ôîðìóëå 5);
10ln
1
)(ln
x
x =
¢
(ïî ôîðìóëå 7).
223. Äèôôåðåíöèðîâàíèå ñóììû, ïðîèçâåäåíèÿ,
÷àñòíîãî. Âî âñåõ ïðèâåäåííûõ íèæå òåîðåìàõ áó-
äåì ñ÷èòàòü çàäàííûå ôóíêöèè u è v äèôôåðåíöè-
ðóåìûìè â òî÷êå õ.
Ò.7.6. Ïðîèçâîäíàÿ ñóììû äâóõ ôóíêöèé âû÷èñëÿ-
åòñÿ ïî ôîðìóëå
vuvu
¢
+
¢
=
¢
+ )(
(òåîðåìà î äèôôåðåíöèðîâàíèè ñóììû).
Ò.7.7. Ïîñòîÿííûé ìíîæèòåëü ìîæíî âûíåñòè çà
çíàê ïðîèçâîäíîé:
uCCu
¢
=
¢
)(
(òåîðåìà î âûíåñåíèè ïîñòîÿííîãî ìíîæèòåëÿ
çà çíàê ïðîèçâîäíîé).
(lg x)
0,6
ÀËÃÅÁÐÀ
§ 23. Ïðîèçâîäíàÿ
ÀËÃÅÁÐÀ
311310
Ðàçäåë VII. ÝËÅÌÅÍÒÛ ÌÀÒ. ÀÍÀËÈÇÀ
Ò.7.8. Ïðîèçâîäíàÿ ïðîèçâåäåíèÿ äâóõ ôóíêöèé âû-
÷èñëÿåòñÿ ïî ôîðìóëå
vuvuuv
¢
+
¢
=
¢
)(
(òåîðåìà î äèôôåðåíöèðîâàíèè ïðîèçâåäåíèÿ).
Ò.7.9. Ïðîèçâîäíàÿ ÷àñòíîãî äâóõ ôóíêöèé âû÷èñ-
ëÿåòñÿ ïî ôîðìóëå
2
v
vuvu
v
u
¢
-
¢
=
¢
÷
ø
ö
ç
è
æ
ïðè óñëîâèè
0)( ¹xv
(òåîðåìà î äèôôåðåíöèðîâàíèè ÷àñòíîãî).
Ï ð è ì å ð 1. Íàéòè ïðîèçâîäíóþ ôóíêöèè:
à)
;5cos7,0sin2 +-= xxy
á)
.log
3
4,0
xx
q à) Èñïîëüçóÿ òåîðåìû 7.6 è 7.7, èìååì
=
¢
+
¢
-+
¢
=
¢
+- 5)cos7,0()sin2()5cos7,0sin2( xxxx
.5)(cos7,0)(sin2
¢
+
¢
-
¢
= xx
Îñòàåòñÿ ïðèìåíèòü ñîîòâåòñòâóþùèå ôîðìóëû
äèôôåðåíöèðîâàíèÿ (ñì. ï. 222). Òîãäà ïîëó÷èì
.sin7,0cos20)sin(7,0cos2 xxxx +=+--
á) Ñîãëàñíî òåîðåìå 7.8, íàõîäèì
.
)
(
loglog
)
(
)
log
(
3
4,0
3
4,0
3
4,0
¢
+
¢
=
¢
xxxxxx
Òåïåðü ïðèìåíèì ôîðìóëû äèôôåðåíöèðîâàíèÿ
è ïîëó÷èì
.
3ln
1ln4,0
3ln
1
log4,0
4,0
4,0
3
6,0
×
+
=×+
-
x
x
x
xxx
n
Íàïðèìåð:
2)32( =
¢
-x
(ïî ôîðìóëå 2);
,2
)
(
1
,10
)
(
32
2
910
--
-=
¢
=
¢
÷
÷
ø
ö
ç
ç
è
æ
=
¢
xx
x
xx
5
2
5
3
5
3
)
(
-
=
¢
xx
(ïî ôîðìóëå 3);
6ln6)6(
xx
=
¢
(ïî ôîðìóëå 5);
10ln
1
)(ln
x
x =
¢
(ïî ôîðìóëå 7).
223. Äèôôåðåíöèðîâàíèå ñóììû, ïðîèçâåäåíèÿ,
÷àñòíîãî. Âî âñåõ ïðèâåäåííûõ íèæå òåîðåìàõ áó-
äåì ñ÷èòàòü çàäàííûå ôóíêöèè u è v äèôôåðåíöè-
ðóåìûìè â òî÷êå õ.
Ò.7.6. Ïðîèçâîäíàÿ ñóììû äâóõ ôóíêöèé âû÷èñëÿ-
åòñÿ ïî ôîðìóëå
vuvu
¢
+
¢
=
¢
+ )(
(òåîðåìà î äèôôåðåíöèðîâàíèè ñóììû).
Ò.7.7. Ïîñòîÿííûé ìíîæèòåëü ìîæíî âûíåñòè çà
çíàê ïðîèçâîäíîé:
uCCu
¢
=
¢
)(
(òåîðåìà î âûíåñåíèè ïîñòîÿííîãî ìíîæèòåëÿ
çà çíàê ïðîèçâîäíîé).
(lg x)
0,6
ÀËÃÅÁÐÀ
§ 23. Ïðîèçâîäíàÿ
ÀËÃÅÁÐÀ
313312
Ðàçäåë VII. ÝËÅÌÅÍÒÛ ÌÀÒ. ÀÍÀËÈÇÀ
Ï ð è ì å ð 1. Èç êàêèõ ôóíêöèé ñîñòàâëåíà
ñëîæíàÿ ôóíêöèÿ
?)12(tg
5
+= xy
q Ýòà ôóíêöèÿ ñîñòîèò èç òðåõ ôóíêöèé:
g(x) = 2x + 1, h(u) = tg u, f(z) = z
5
. Â ñàìîì äåëå,
f (h (g (x))) = (tg (g (x)))
5
= (tg (2x + 1))
5
= tg
5
(2x + 1). n
Ïóñòü
))(( xgfy =
ñëîæíàÿ ôóíêöèÿ, ïðè÷åì
ôóíêöèÿ
)(xgu =
äèôôåðåíöèðóåìà â òî÷êå õ, à
ôóíêöèÿ
)(ufy =
äèôôåðåíöèðóåìà â ñîîòâåòñòâó-
þùåé òî÷êå
.u
Òîãäà ôóíêöèÿ
))(( xgfy =
äèôôå-
ðåíöèðóåìà â òî÷êå õ, ïðè÷åì
).())(( xgxgfy
¢
×
¢
=
¢
Çàïèñü
))(( xgf
¢
îçíà÷àåò, ÷òî ïðîèçâîäíàÿ âû÷èñ-
ëÿåòñÿ ïî ôîðìóëå äëÿ
),(xf
¢
íî âìåñòî õ íóæíî ïîä-
ñòàâèòü
).(xg
Ï ð è ì å ð 2. Íàéòè
.))53((
4
¢
+x
q Çäåñü
+==+= xxgfuufxxg 3())((,)(,53)(
4
.)5
4
+
Çíà÷èò,
=
¢
+×+=
¢
×
¢
=
¢
)53()53(4)())((
3
xxxgxgfy
.)53(123)53(4
33
+=×+= xx
n
Îòìåòèì ÷àñòíûé ñëó÷àé ïðàâèëà äèôôåðåíöè-
ðîâàíèÿ ñëîæíîé ôóíêöèè:
).())(( bkxfkbkxf +
¢
=
¢
+
Òàê,
.
752
7
)
75
(
;3cos3
)
3
(
sin
x
xxx
-
-=
¢
-=
¢
Ï ð è ì å ð 2. Âû÷èñëèòü
),0(f
¢
åñëè
.
1
2
)
(
2
+
=
x
xf
x
q Ñíà÷àëà íàéäåì
).(xf
¢
Âîñïîëüçîâàâøèñü òåî-
ðåìîé 7.9, ïîëó÷èì
=
+
¢
+-+
¢
=
¢
22
22
)1(
)1(2)1()2(
)(
x
xx
xf
xx
.
)1(
22)1(2ln2
22
2
+
×-+×
=
x
xx
xx
Òåïåðü âû÷èñëèì
.2ln
)10(
022)10(2ln2
)0(
22
020
=
+
××-+×
=
¢
f
n
224. Ñëîæíàÿ ôóíêöèÿ è åå äèôôåðåíöèðîâà-
íèå. Ðàññìîòðèì ôóíêöèþ
.sin
2
xy =
×òîáû íàéòè
çíà÷åíèå ýòîé ôóíêöèè â ôèêñèðîâàííîé òî÷êå õ,
íóæíî: 1) âû÷èñëèòü õ
2
; 2) íàéòè çíà÷åíèå ñèíóñà
ïðè ïîëó÷åííîì çíà÷åíèè õ
2
. Èíûìè ñëîâàìè, ñíà-
÷àëà íàäî íàéòè çíà÷åíèå ôóíêöèè
,)(
2
xxg =
à çà-
òåì íàéòè
).(sin xg
 ïîäîáíûõ ñëó÷àÿõ ãîâîðÿò, ÷òî
çàäàíà ñëîæíàÿ ôóíêöèÿ
)).(( xgf
 äàííîì ñëó÷àå
,)(
2
xxgu ==
à
.sin)( uuf =
ÀËÃÅÁÐÀ
§ 23. Ïðîèçâîäíàÿ
ÀËÃÅÁÐÀ
313312
Ðàçäåë VII. ÝËÅÌÅÍÒÛ ÌÀÒ. ÀÍÀËÈÇÀ
Ï ð è ì å ð 1. Èç êàêèõ ôóíêöèé ñîñòàâëåíà
ñëîæíàÿ ôóíêöèÿ
?)12(tg
5
+= xy
q Ýòà ôóíêöèÿ ñîñòîèò èç òðåõ ôóíêöèé:
g(x) = 2x + 1, h(u) = tg u, f(z) = z
5
. Â ñàìîì äåëå,
f (h (g (x))) = (tg (g (x)))
5
= (tg (2x + 1))
5
= tg
5
(2x + 1). n
Ïóñòü
))(( xgfy =
ñëîæíàÿ ôóíêöèÿ, ïðè÷åì
ôóíêöèÿ
)(xgu =
äèôôåðåíöèðóåìà â òî÷êå õ, à
ôóíêöèÿ
)(ufy =
äèôôåðåíöèðóåìà â ñîîòâåòñòâó-
þùåé òî÷êå
.u
Òîãäà ôóíêöèÿ
))(( xgfy =
äèôôå-
ðåíöèðóåìà â òî÷êå õ, ïðè÷åì
).())(( xgxgfy
¢
×
¢
=
¢
Çàïèñü
))(( xgf
¢
îçíà÷àåò, ÷òî ïðîèçâîäíàÿ âû÷èñ-
ëÿåòñÿ ïî ôîðìóëå äëÿ
),(xf
¢
íî âìåñòî õ íóæíî ïîä-
ñòàâèòü
).(xg
Ï ð è ì å ð 2. Íàéòè
.))53((
4
¢
+x
q Çäåñü
+==+= xxgfuufxxg 3())((,)(,53)(
4
.)5
4
+
Çíà÷èò,
=
¢
+×+=
¢
×
¢
=
¢
)53()53(4)())((
3
xxxgxgfy
.)53(123)53(4
33
+=×+= xx
n
Îòìåòèì ÷àñòíûé ñëó÷àé ïðàâèëà äèôôåðåíöè-
ðîâàíèÿ ñëîæíîé ôóíêöèè:
).())(( bkxfkbkxf +
¢
=
¢
+
Òàê,
.
752
7
)
75
(
;3cos3
)
3
(
sin
x
xxx
-
-=
¢
-=
¢
Ï ð è ì å ð 2. Âû÷èñëèòü
),0(f
¢
åñëè
.
1
2
)
(
2
+
=
x
xf
x
q Ñíà÷àëà íàéäåì
).(xf
¢
Âîñïîëüçîâàâøèñü òåî-
ðåìîé 7.9, ïîëó÷èì
=
+
¢
+-+
¢
=
¢
22
22
)1(
)1(2)1()2(
)(
x
xx
xf
xx
.
)1(
22)1(2ln2
22
2
+
×-+×
=
x
xx
xx
Òåïåðü âû÷èñëèì
.2ln
)10(
022)10(2ln2
)0(
22
020
=
+
××-+×
=
¢
f
n
224. Ñëîæíàÿ ôóíêöèÿ è åå äèôôåðåíöèðîâà-
íèå. Ðàññìîòðèì ôóíêöèþ
.sin
2
xy =
×òîáû íàéòè
çíà÷åíèå ýòîé ôóíêöèè â ôèêñèðîâàííîé òî÷êå õ,
íóæíî: 1) âû÷èñëèòü õ
2
; 2) íàéòè çíà÷åíèå ñèíóñà
ïðè ïîëó÷åííîì çíà÷åíèè õ
2
. Èíûìè ñëîâàìè, ñíà-
÷àëà íàäî íàéòè çíà÷åíèå ôóíêöèè
,)(
2
xxg =
à çà-
òåì íàéòè
).(sin xg
 ïîäîáíûõ ñëó÷àÿõ ãîâîðÿò, ÷òî
çàäàíà ñëîæíàÿ ôóíêöèÿ
)).(( xgf
 äàííîì ñëó÷àå
,)(
2
xxgu ==
à
.sin)( uuf =