ЛЕКЦІЇ ² ВПРАВИ

Ïðèêëàä 7.7.2. y = x sin x, x∈R.
Ðîçâ’ÿçàííÿ.
( )′ ( )′
y′ = x ⋅ sin x+ x sin x = sin x+ x⋅ cosx.
sinx
Ïðèêëàä 7.7.3. y = .
x
Ðîçâ’ÿçàííÿ.
( sin x) ′ ⋅x−sin x⋅( x)
′
cos x⋅x−sin
x
y′ = = .
2 2
x
x
π
Ïðèêëàä 7.7.4. y = tg x, x≠ +πn,
n∈Z . Òðåáà çíàéòè y′.
2
Ðîçâ’ÿçàííÿ.
( )
⎛sin x ⎞ cos cos sin sin 1
( tg )
′ x⋅ x− x⋅ − x
y′ = x ′ = ⎜
2 2
cos x
⎟ = = . (7.7.1)
⎝ ⎠
cos x cos x
Ïðèêëàä 7.7.5. y = ctg x, x ≠πn,
n∈Z . Òðåáà çíàéòè y′.
Ðîçâ’ÿçàííÿ.
( )
⎛cos x ⎞ sin sin cos cos 1
( ctg )
′ − x⋅ x − x⋅
x
y′ = x ′ = ⎜
2 2
sin x
⎟ = = − . (7.7.2)
⎝ ⎠
sin x sin x
Çàóâàæèìî, ùî îñòàíí³ äâà ïðèêëàäè º òåîðåòè÷íèìè.
7.8. ÏÎÕ²ÄÍÀ ÑÊËÀÄÅÍί ÔÓÍÊÖ²¯
7.8.1. Òåîðåìà (ïðî ïîõ³äíó ñêëàäåíî¿ ôóíêö³¿)
ßêùî ôóíêö³ÿ ó = f(u) ìຠïîõ³äíó â òî÷ö³ u, à u º ôóíêö³ÿ
â³ä x ³ u(õ) ìຠïîõ³äíó â òî÷ö³ õ, òî ñêëàäåíà ôóíêö³ÿ
ó = f(u(x)) ìຠïîõ³äíó â òî÷ö³ õ, ïðè÷îìó
dy dy du
= ⋅ àáî y x
′ = y u
′ ⋅ u x
′. (7.8.1)
dx du dx
Ä î â å ä å í í ÿ. Îñê³ëüêè ôóíêö³ÿ ó = f(u) ìຠïîõ³äíó â
òî÷ö³ u, òî
∆y
lim = y′
u .
∆u
∆u→0
Çà âèçíà÷åííÿì ãðàíèö³ ôóíêö³¿ ð³çíèöÿ ì³æ ¿¿ çíà÷åííÿì
³ ãðàíèöåþ ìîæå áóòè îòðèìàíà ÿê çàâãîäíî ìàëîþ,
òîáòî
∆y
= y′
u
+α( u,
∆u)
,
∆u
∆ y = y ∆ u+α u ∆u ∆ u. Àíà-
äå α( u, ∆u)
→ 0 ïðè ∆u
→ 0 . Îòæå, ′
u ( , )
ëîã³÷íî ∆ u = u′
∆ x+β( x,
∆x)
∆ x , äå β( x,
∆ )
x
x ïðÿìóº äî íóëÿ ïðè
∆x → 0. ϳäñòàâëÿþ÷è âèðàç äëÿ ∆u â ∆ó, îòðèìàºìî
∆ y = y′ u′ + y′ β∆ x + u′
α∆ x + αβ∆ x.
u x u x
Ðîçä³ëèâøè îáèäâ³ ÷àñòèíè îñòàííüî¿ ð³âíîñò³ íà ∆õ ≠ 0 ³
ïåðåéøîâøè äî ãðàíèö³ ïðè ∆õ → 0, îòðèìàºìî
∆y lim = y ′
u⋅
u ′
x. (7.8.2)
∆x
∆x→0
Îòæå, çã³äíî ç îçíà÷åííÿì ïîõ³äíî¿ ³ ñï³ââ³äíîøåííÿì
(7.8.2) ñïðàâåäëèâà ôîðìóëà (7.8.1).
Ôîðìóëà (7.8.1) äຠòàêå ïðàâèëî äèôåðåíö³þâàííÿ ñêëàäåíèõ
ôóíêö³é: ïîõ³äíà ñêëàäåíî¿ ôóíêö³¿ äîð³âíþº äîáóòêó
ïîõ³äíî¿ çîâí³øíüî¿ ôóíêö³¿ íà ïîõ³äíó âíóòð³øíüî¿
ôóíêö³¿.
Ç à ó â à æ å í í ÿ. ßêùî ó = f(u), u = ϕ(v), v = g(x), òî
ó = f(ϕ(g(x))), y′ x
= y′ u
⋅u′ v
⋅ v′
x
.
7.8.2. Òàáëèöÿ ïîõ³äíèõ ñêëàäåíèõ ôóíêö³é
Íà îñíîâ³ ôîðìóë (7.4.1) — (7.4.8), (7.5.8) — (7.5.11),
(7.7.1) — (7.7.2) ìîæíà áóëî ñêëàñòè òàáëèöþ ïîõ³äíèõ
îñíîâíèõ åëåìåíòàðíèõ ôóíêö³é. Àëå ìè öüîãî íå áóäåìî
ðîáèòè ÷åðåç òå, ùî çàâäÿêè òåîðåì³ 7.8.1 ³ íà îñíîâ³ òèõ
æå ôîðìóë (7.4.1) — (7.4.8), (7.5.8) — (7.5.11), (7.7.1) —
(7.7.2) ìîæíà ñêëàñòè á³ëüø çàãàëüíó òàáëèöþ. ¯¿ ìè íàçâåìî
òàáëèöåþ ïîõ³äíèõ ñêëàäåíèõ ôóíêö³é. Íàâåäåìî ¿¿.
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