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 ÿêîñò³ ïðèêëàäà ðîçãëÿíåìî âèïàäîê, êîëè<br />
¢ ¢<br />
( ) ( )<br />
1<br />
α = . Òîä³<br />
2<br />
1 1 -1<br />
1<br />
2 2<br />
x = x = x = . (7.4.6)<br />
2 2 x<br />
Ôîðìóëó (7.4.6) ñë³ä çàïàì’ÿòàòè îêðåìî, îñê³ëüêè ¿¿<br />
÷àñòî âèêîðèñòîâóþòü.<br />
7.4.5. Ïîõ³äíà ôóíêö³¿ y=sinx, x∈ R<br />
y+D y= sin x+Dx , D x¹ 0 .<br />
1 0 . ( )<br />
Dx<br />
x<br />
y sin x x sin x 2sin cos<br />
æ D<br />
D = +D - = ç x<br />
ö<br />
+ 2 çè 2 ø ÷ .<br />
2 0 . ( )<br />
3 0 .<br />
4 0 .<br />
Dx<br />
æ D xö<br />
2sin × cos<br />
ç x+ D y 2 çè 2 ø÷<br />
=<br />
.<br />
Dx<br />
Dx<br />
∆ y 1<br />
lim = 2 ⋅ ⋅ cosx<br />
= cosx.<br />
∆x<br />
2<br />
∆x→0<br />
Çàóâàæèìî, ùî ïðè îá÷èñëåíí³ ãðàíèö³ ìè âèêîðèñòàëè<br />
òåîðåìó ïðî äîáóòîê ãðàíèöü, íåïåðåðâí³ñòü ôóíêö³¿<br />
y = cosx ³ ôîðìóëó (6.3.8). Îòæå,<br />
(sin x)′ = cos x. (7.4.7)<br />
7.4.6. Ïîõ³äíà ôóíêö³¿ y = cos x, x∈R<br />
Àíàëîã³÷íî äîâîäèòüñÿ, ùî äëÿ ôóíêö³¿ y = cos x ó äîâ³ëüí³é<br />
òî÷ö³ x∈R ³ñíóº ïîõ³äíà ³ äîð³âíþº –sin x, òîáòî<br />
(cos x)′ = −sin x. (7.4.8)<br />
Ïðîïîíóºìî ÷èòà÷åâ³ ôîðìóëó (7.4.8) âèâåñòè ñàìîñò³éíî.<br />
7.5. ÏÎÕ²ÄÍÀ ÎÁÅÐÍÅÍί ÔÓÍÊÖ²¯<br />
Ïåðø í³æ âèâåñòè ôîðìóëè ïîõ³äíèõ îáåðíåíèõ ôóíêö³é,<br />
äîâåäåìî òåîðåìó ïðî ïîõ³äíó îáåðíåíî¿ ôóíêö³¿.<br />
Òåîðåìà 7.5.1 (ïðî ïîõ³äíó îáåðíåíî¿ ôóíêö³¿). Íåõàé<br />
ôóíêö³ÿ y = f(x) çàäîâîëüíÿº âñ³ óìîâè òåîðåìè ïðî ³ñíóâàííÿ<br />
îáåðíåíî¿ ôóíêö³¿ (äèâ. òåîð. 6.3.8) ³ â òî÷ö³ õ 0 ìàº<br />
ïîõ³äíó f′(x 0 ) ≠ 0. Òîä³ îáåðíåíà äëÿ íå¿ ôóíêö³ÿ x = ϕ(y) ó<br />
òî÷ö³ ìຠy 0 = f(x 0 ) òàêîæ ïîõ³äíó ³ ñïðàâåäëèâà ôîðìóëà<br />
1<br />
ϕ ¢ ( y0)<br />
=<br />
f ¢ ( x )<br />
. (7.5.1)<br />
Äîâåäåííÿ. Íàäàìî òî÷ö³ y 0 ïðèðîñòó ∆y ≠ 0. Òîä³ îáåðíåíà<br />
ôóíêö³ÿ x = ϕ(y) ó òî÷ö³ õ 0 ä³ñòàíå ïðèð³ñò ∆õ, ïðè÷îìó<br />
âíàñë³äîê ¿¿ ñòðîãî¿ ìîíîòîííîñò³ ìàòèìåìî ∆x ≠ 0. Ó<br />
öüîìó âèïàäêó ñïðàâäæóºòüñÿ òîòîæí³ñòü<br />
0<br />
D x 1 = D y D y .<br />
Dx<br />
Ïåðåéäåìî â ö³é ð³âíîñò³ äî ãðàíèö³ ïðè D y ® 0 . Âíàñë³äîê<br />
íåïåðåðâíîñò³ îáåðíåíî¿ ôóíêö³¿ ∆õ òàêîæ áóäå ïðÿìóâàòè<br />
äî íóëÿ.<br />
Òàêèì ÷èíîì,<br />
∆ x 1 1<br />
lim = lim =<br />
∆y<br />
∆ y f′<br />
( x0)<br />
. (7.5.2)<br />
∆x<br />
∆y→0 ∆x→0<br />
Ïðè âèâåäåíí³ ôîðìóëè (7.5.2) ìè ñêîðèñòàëèñÿ òåîðåìîþ<br />
ïðî ãðàíèöþ ÷àñòêè, à òàêîæ óìîâîþ òåîðåìè<br />
(f′(x 0 ) ≠ 0).<br />
Òàêèì ÷èíîì, ³ñíóº ïîõ³äíà îáåðíåíî¿ ôóíêö³¿ x = ϕ(y) ó<br />
òî÷ö³ y 0 = f(x 0 ) ³ ñïðàâåäëèâà ôîðìóëà (7.5.1).<br />
Òåîðåìó äîâåäåíî.<br />
Çàóâàæåííÿ. Ôîðìóëà (7.5.1) ïîâ’ÿçóº ïðè ïåâíèõ<br />
óìîâàõ çíà÷åííÿ ïîõ³äíèõ ïðÿìî¿ ³ îáåðíåíî¿ ôóíêö³é ó<br />
â³äïîâ³äíèõ òî÷êàõ. ßêùî æ òåïåð ôóíêö³ÿ y = f(x) ìàº<br />
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