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Òàáëèöÿ ïîõ³äíèõ ñêëàäåíèõ ôóíêö³é<br />
y y′ y y′<br />
1 c = const 0 9 sin u(x) u′(x) cos u(x)<br />
m<br />
2 u ( x),<br />
m∈<br />
R<br />
3 ux ( )<br />
u( x<br />
4 a ) ( a > 0, a ≠1)<br />
5<br />
u( x)<br />
e<br />
6<br />
log<br />
a<br />
ux ( )<br />
( a > 0, a ≠1)<br />
7 lnu(x)<br />
8<br />
( )<br />
⎡⎣ux<br />
( ) ⎤⎦<br />
v x<br />
m− mu<br />
1 ( x) u′ ( x)<br />
10 cos ux ( ) −u′<br />
( x)sin u( x)<br />
u′<br />
( x)<br />
2 ux ( )<br />
u′<br />
( x)<br />
11 tg u(x) 2<br />
cos ux ( )<br />
u′<br />
( x)<br />
u( x<br />
a<br />
) ln a⋅u′<br />
( x)<br />
12 ctg u(x) −<br />
2<br />
sin ux ( )<br />
e<br />
u( x)<br />
u′<br />
( x)<br />
u′ ( x)<br />
13 arcsin u(x) 2<br />
1 − u ( x)<br />
u′<br />
( x)<br />
ux ( )lna<br />
u′<br />
( x)<br />
ux ( )<br />
v<br />
v 1<br />
uv u vu − u<br />
u′<br />
( x)<br />
14 arccos u(x)<br />
−<br />
2<br />
1 − u ( x)<br />
u′<br />
( x)<br />
15 arctg u(x) 2<br />
1 + u ( x)<br />
u′<br />
( x)<br />
′ ln + ′ 16 arcctg u(x) −<br />
2<br />
1 + u ( x)<br />
Çàóâàæåííÿ 1. Ïîõ³äíà, ÿêà çàïèñàíà ó ñüîìîìó ðÿäêó<br />
òàáëèö³, íàçèâàºòüñÿ ëîãàðèôì³÷íîþ ïîõ³äíîþ. Ëîãàðèôì³÷íó<br />
ïîõ³äíó ( ln u)<br />
′ ′ = íàçèâàþòü òàêîæ â³äíîñíîþ øâèä-<br />
u<br />
u<br />
ê³ñòþ çì³íè ôóíêö³¿, àáî òåìïîì çì³íè ôóíêö³¿. Ñë³ä ñêàçàòè,<br />
ùî îñòàíí³é òåðì³í äóæå ÷àñòî âèêîðèñòîâóºòüñÿ ó<br />
òåîðåòè÷íèõ ïèòàííÿõ ç åêîíîì³êè.<br />
Çàóâàæåííÿ 2. Íàâåäåíîþ òàáëèöåþ ìîæíà êîðèñòóâàòèñÿ<br />
³ äëÿ âèçíà÷åííÿ ïîõ³äíèõ â³ä îñíîâíèõ åëåìåíòàðíèõ<br />
ôóíêö³é, ÿê³ çàëåæàòü ò³ëüêè â³ä õ. Äëÿ öüîãî ó íàâåäåí³é<br />
òàáëèö³ òðåáà ïîêëàñòè u(x) =x (u′ = 1).<br />
Ïðèêëàäè 7.7.6 — 7.7.10. Êîðèñòóþ÷èñü ôîðìóëàìè äèôåðåíö³þâàííÿ,<br />
çíàéòè ïîõ³äí³ òàêèõ ôóíêö³é:<br />
7.7.6.<br />
2 1 3 2<br />
y = x + + x +<br />
2 ;<br />
x 5<br />
1 2+<br />
x x arcsinx x<br />
7.7.7. y = sin2x+ ln −2 ⋅ cos3 x+ + arctg + arccose<br />
2 2− x tgx x+<br />
2<br />
2x<br />
7.7.8. y = x ; 7.7.9.<br />
7.7.10.<br />
cos αtg<br />
α<br />
y( α ) =<br />
1+<br />
2tg<br />
c<br />
.<br />
5<br />
yt () =<br />
;<br />
asin<br />
t − bcos<br />
t<br />
Ðîçâ’ÿçàííÿ.<br />
7.7.6. Çàïèøåìî ôóíêö³þ ó âèãëÿä³:<br />
1<br />
2 −2 2<br />
3<br />
y= x + x + x + .<br />
5<br />
Çàñòîñîâóþ÷è ôîðìóëó äèôåðåíö³þâàííÿ ñòåïåíåâî¿ ôóíêö³¿<br />
³ ïðàâèëî äèôåðåíö³þâàííÿ àëãåáðà¿÷íî¿ ñóìè, áóäåìî<br />
ìàòè:<br />
2<br />
−3 1<br />
3<br />
y′ = 2x − 2x + x −<br />
.<br />
3<br />
7.7.7. Ñïî÷àòêó çíàéäåìî ïîõ³äí³ êîæíîãî ³ç àëãåáðè÷íèõ<br />
äîäàíê³â, ÿê³ âõîäÿòü ó ñòðóêòóðó äàíî¿ ôóíêö³¿:<br />
⎛1 ⎞<br />
′ 1<br />
⎜ sin 2x⎟<br />
= ⋅ 2cos 2x = cos 2x<br />
,<br />
⎝2 ⎠ 2<br />
⎛<br />
′<br />
2+ x ⎞ 1⎛ 2+<br />
x⎞ 1<br />
ln ln<br />
⎛<br />
( ln ( 2 ) ln ( 2<br />
′ ⎞<br />
= = + − − ))<br />
=<br />
⎜<br />
⎜ ⎟ ⎜ ⎟<br />
2 ⎟<br />
x x<br />
⎝ −x<br />
⎠ 2⎝<br />
2−x⎠<br />
2⎝ ⎠<br />
2x<br />
;<br />
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