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ïîõ³äíó â äîâ³ëüí³é òî÷ö³ õ ³ f′(x) ≠ 0, òî ôîðìóëà âèêîíó-<br />
ºòüñÿ äëÿ òî÷îê õ:<br />
àáî<br />
1<br />
ϕ ¢ ( y)<br />
= ,<br />
f ¢<br />
(7.5.3)<br />
( x)<br />
1<br />
x′ y<br />
=<br />
y ′<br />
. (7.5.4)<br />
Ôîðìóëà (7.5.4) çðó÷íà äëÿ êîðèñòóâàííÿ ó äàí³é ñèòóàö³¿.<br />
Âîíà êîíêðåòíî (ïî íèæíüîìó ³íäåêñó) ïîêàçóº ïî<br />
ÿê³é çì³íí³é çíàõîäèòüñÿ ïîõ³äíà.<br />
²ç ôîðìóëè (7.5.4) ëåãêî îòðèìàòè ³ òàêó:<br />
x<br />
1<br />
y ′ x<br />
= , xy<br />
0<br />
x<br />
′ ≠ . (7.5.5)<br />
′<br />
y<br />
7.5.1. Ïîõ³äí³ îáåðíåíèõ òðèãîíîìåòðè÷íèõ ôóíêö³é<br />
Ïî÷íåìî ç ôóíêö³¿ y = arcsin x, x∈(−1, 1). Çà îçíà÷åííÿì<br />
îáåðíåíî¿ ôóíêö³¿ y = arcsin x (äèâ. äîä. 2) ìàºìî<br />
⎛ π π⎞<br />
x = sin y, y∈ ⎜ − ;<br />
2 2 ⎟<br />
⎝ ⎠ . (7.5.6)<br />
Òåïåð äëÿ çíàõîäæåííÿ ïîõ³äíî¿ îáåðíåíî¿ ôóíêö³¿<br />
y = arcsin x çàñòîñóºìî ôîðìóëó (7.5.5).  ðåçóëüòàò³ îòðèìàºìî<br />
1 1<br />
y′ x<br />
= =<br />
x′ cos y<br />
. (7.5.7)<br />
y<br />
Ôîðìóëà (7.5.7) âèçíà÷ຠïîõ³äíó îáåðíåíî¿ ôóíêö³¿<br />
y = arcsin x ó íåÿâí³é ôîðì³. Äëÿ çîáðàæåííÿ ¿¿ â ÿâí³é<br />
⎛ π π⎞<br />
ôîðì³ âèðàçèìî cos y ÷åðåç õ. Îñê³ëüêè y ∈ ⎜ − ;<br />
2 2 ⎟<br />
⎝ ⎠ , òî<br />
ôóíêö³ÿ cos y íàáóâຠò³ëüêè äîäàòíèõ çíà÷åíü ³ ¿¿ ìîæíà<br />
çàïèñàòè ó âèãëÿä³:<br />
cos y 1 sin y 1 x<br />
2 2<br />
= − = − .<br />
ßêùî òåïåð îñòàííþ ôîðìóëó ï³äñòàâèìî â (7.5.7) ³ âðàõóºìî,<br />
ùî y = arcsin x, òî îñòàòî÷íî îòðèìàºìî ôîðìóëó<br />
′ 1 =<br />
1− x<br />
( arcsin x) 2<br />
. (7.5.8)<br />
Àíàëîã³÷íî ìîæíà âèâåñòè ôîðìóëó ïîõ³äíèõ äëÿ îáåðíåíèõ<br />
òðèãîíîìåòðè÷íèõ ôóíêö³é arccos x, arctg x, arcctg x.<br />
Äëÿ íèõ ñïðàâåäëèâ³ òàê³ ôîðìóëè:<br />
′ 1 =−<br />
1− x<br />
( arccos x) 2<br />
( arctg x) 2<br />
, (7.5.9)<br />
′ 1 =<br />
1 + x , (7.5.10)<br />
′ 1 =−<br />
1 + x . (7.5.11)<br />
( arcctg x) 2<br />
Ôîðìóëè (7.5.9) – (7.5.11) ïðîïîíóºìî ÷èòà÷åâ³ âèâåñòè<br />
ñàìîñò³éíî.<br />
7.6. ÎÑÍÎÂͲ ÏÐÀÂÈËÀ<br />
ÄÈÔÅÐÅÍÖ²ÞÂÀÍÍß ÔÓÍÊÖ²É<br />
Ðîçãëÿíåìî ôóíêö³¿ u = sin x, v = x ³ ïîñòàâèìî çàäà÷ó<br />
ïðî çíàõîäæåííÿ ïîõ³äíèõ òàêèõ ôóíêö³é: y = sin x ± x,<br />
sinx<br />
y = x sin x, y = . Êîðèñòóþ÷èñü çàãàëüíèì ïðàâèëîì çíàõîäæåííÿ<br />
ïîõ³äíèõ, öþ ïðîáëåìó ìîæíà ðîçâ’ÿçàòè àëå äî-<br />
x<br />
ñèòü ãðîì³çäêî. Êðàùå ïîñòàâëåíó çàäà÷ó ðîçâ’ÿçàòè çà<br />
äîïîìîãîþ ïðàâèë äèôåðåíö³þâàííÿ ôóíêö³é, ÿê³ ó öüîìó<br />
ïóíêò³ áóäå ïîäàíî ó âèãëÿä³ òåîðåì.<br />
Òåîðåìà 7.6.1 (ïðî äèôåðåíö³þâàííÿ ñóìè ³ ð³çíèö³).<br />
Íåõàé ôóíêö³¿ u(x) i v(x) äèôåðåíö³éîâí³ íà ³íòåðâàë³<br />
(a, b). Òîä³ ôóíêö³¿ y = u(x) ±v(x) òàêîæ äèôåðåíö³éîâí³ íà<br />
³íòåðâàë³ (a, b) ³ ñïðàâåäëèâ³ ôîðìóëè<br />
( )′<br />
y′ = u ± v = u′ ± v′<br />
. (7.6.1)<br />
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