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Òåîðåìà 7.6.2 (ïðî äèôåðåíö³þâàííÿ äîáóòêó). Íåõàé<br />
ôóíêö³¿ u(x) i v(x) äèôåðåíö³éîâí³ íà ³íòåðâàë³ (a, b). Òîä³<br />
ôóíêö³ÿ y = u(x)v(x) òàêîæ äèôåðåíö³éîâíà íà ³íòåðâàë³<br />
(a, b) ³ ñïðàâåäëèâà ôîðìóëà<br />
( )′<br />
y′ = uv = uv ′ + v′<br />
u . (7.6.2)<br />
Íàñë³äîê. Ïðè äèôåðåíö³þâàíí³ (ÿêùî öå ìîæëèâî)<br />
ñòàëó ìîæíà âèíîñèòè çà çíàê ïîõ³äíî¿, òîáòî ñïðàâäæóºòüñÿ<br />
ôîðìóëà<br />
( )′<br />
y′ = cu = cu′<br />
. (7.6.3)<br />
Äîâåäåííÿ. Ó ôîðìóë³ (7.6.3) ïðèïóñòèìî, ùî u(x)<br />
äîâ³ëüíà äèôåðåíö³éîâíà íà ³íòåðâàë³ (a, b) ôóíêö³ÿ, à<br />
v(x) =ñ = const. Îñê³ëüêè v′ = (c)′ = 0, òî ³ç ôîðìóëè (7.6.2)<br />
ä³éñíî âèïëèâຠôîðìóëà (7.6.3).<br />
Çàóâàæèìî, ùî ³ç òåîðåìè 7.6.1 ³ íàñë³äêó òåîðåìè 7.6.2<br />
âèïëèâຠòàêå òâåðäæåííÿ:<br />
ë³í³éíà êîìá³íàö³ÿ äèôåðåíö³éîâíèõ íà ³íòåðâàë³ (a, b).<br />
ôóíêö³é (f i (x), i =1,2,…,n)<br />
ìຠïîõ³äíó<br />
n<br />
x ( ) = cf( x) + cf( x) + ... + cf( x) = ∑ cf( x)<br />
11 22<br />
n n<br />
i=<br />
1<br />
i i<br />
i=<br />
1<br />
′ ( x) = c11 f′<br />
( x) + c22<br />
f′ ( x) + ... + cnf′ n( x) = ∑ cf′<br />
i i( x), x∈( a, b)<br />
. (7.6.4)<br />
Òåîðåìà 7.6.3 (ïðî äèôåðåíö³þâàííÿ ÷àñòêè). Íåõàé<br />
ôóíêö³¿ u(x) i v(x) äèôåðåíö³éîâí³ íà ³íòåðâàë³ (a, b) ³<br />
ux ( )<br />
êð³ì òîãî v(x) ≠ 0∀x∈(a, b). Òîä³ ôóíêö³ÿ y = òàêîæ<br />
vx ( )<br />
äèôåðåíö³éîâíà íà ³íòåðâàë³ (a, b) ³ ñïðàâåäëèâà ôîðìóëà<br />
n<br />
⎛u⎞<br />
′ uv ′ − vu ′<br />
y′ = ⎜ =<br />
2<br />
v<br />
⎟<br />
. (7.6.5)<br />
⎝ ⎠ v<br />
Äîâåäåìî îäíó ³ç òåîðåì 7.6.1 — 7.6.3, íàïðèêëàä 7.6.2<br />
(äîâåäåííÿ ³íøèõ òåîðåì ïðîïîíóºìî çä³éñíèòè ÷èòà÷åâ³<br />
ñàìîñò³éíî).<br />
Äîâåäåííÿ. Éîãî áóäåìî ïðîâîäèòè çà â³äîìîþ óæå<br />
ñõåìîþ.<br />
1 0 . y +∆ y = ( u +∆ u)( v +∆ v)<br />
.<br />
2 0 . ∆ y= ( u+∆ u)( v+∆v)<br />
− uv=∆u⋅ v+ u⋅∆ v+∆u⋅∆ v.<br />
3 0 .<br />
4 0 .<br />
∆y ∆u ∆v ∆u<br />
= ⋅ v+ ⋅ u+ ⋅∆v.<br />
∆x ∆x ∆x ∆x<br />
∆y lim = u ′ ⋅ v + v ′ ⋅ u .<br />
∆x<br />
∆→ x 0<br />
Ïðè äîâåäåíí³ ö³º¿ ð³âíîñò³ ìè ñêîðèñòàëèñÿ îçíà÷åííÿì<br />
ïîõ³äíî¿, òåîðåìîþ ïðî ãðàíèöþ ñóìè, à òàêîæ òåîðåìîþ<br />
ïðî íåïåðåðâí³ñòü äèôåðåíö³éîâíî¿ ôóíêö³¿ v( lim v 0)<br />
∆x→0<br />
∆ = .<br />
Ó â³äïîâ³äíîñò³ äî îçíà÷åííÿ ïîõ³äíî¿ îòðèìàëè ôîðìóëó<br />
(7.6.2).<br />
Îòæå, òåîðåìó äîâåäåíî.<br />
Çàóâàæåííÿ. Äîâåäåííÿ òåîðåì 7.6.1 — 7.6.3 áóëî<br />
çä³éñíåíî çà óìîâè, ùî çì³ííà x∈(a, b). Ó çâ’ÿçêó ç öèì<br />
ñë³ä ñêàçàòè, ùî àíàëîã³÷í³ òåîðåìè ìîæóòü áóòè äîâåäåí³<br />
íà áóäü-ÿê³é ÷èñëîâ³é ìíîæèí³ Õ (ìîæëèâî, ç äåÿêèìè îáìåæåííÿìè).<br />
Çîêðåìà, ÿêùî äèôåðåíö³éîâí³ ôóíêö³¿ çàäàí³<br />
íà ñåãìåíò³ [a, b], òî òðåáà ðîçãëÿäàòè îäíîñòîðîíí³ ïîõ³äí³<br />
â òî÷êàõ à ³ b.<br />
7.7. ÏÐÈÊËÀÄÈ ÇÀÑÒÎÑÓÂÀÍÍß ÎÑÍÎÂ-<br />
ÍÈÕ ÏÐÀÂÈË ÄÈÔÅÐÅÍÖÞÂÀÍÍß<br />
ÔÓÍÊÖ²É<br />
Ïðèêëàä 7.7.1. y = sin x + x, x∈R.<br />
Ðîçâ’ÿçàííÿ.<br />
( )′ ( )′<br />
y′ = sin x + x = cos x + 1 .<br />
212 213
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