Tšimoloho ea Mosebetsi: Khopolo-taba, Tšebeliso le Palo
Tšimoloho ea mosebetsi ke khopolo ea motheo ho calculus, e nang le lits'ebetso tse ngata mafapheng a fapaneng a mahlale, joalo ka fisiks, moruo, baeloji le boenjiniere. Ka ho utloisisa ts'imoloho, re ka sekaseka hore na ts'ebetso e fetoha joang ha boleng ba liphetoho tsa eona tse ikemetseng tsa phetoho. Sehloohong sena, re tla akaretsa metheo ea ts'ebetso, melao e meng ea bohlokoa, le lits'ebetso tse ling tsa lefats'e la nnete.
Tlhaloso ea Li-derivatives
Sephetho sa mosebetsi ntlheng ke sekgahla sa phetoho ya boleng ba mosebetsi mabapi le boleng ba phetoho e ikemetseng ntlheng eo. Ka molao, haeba \( f(x) \) e le mosebetsi, jwale sephetho sa \( f \) ho \( x = a \) se bontshwa ke \( f'(a) \) kapa \( \frac{d}{dx} f(x) \bigg|_{x=a} \). Tlhaloso e hlaloswa e le moedi:
\[ f'(a) = \lim_{\Delta x \to 0} \frac{f(a + \Delta x) – f(a)}{\Delta x} \]
Mona, \( \Delta x \) ke phetoho e nyane ho \( x \), mme \( f(a + \Delta x) – f(a) \) ke phetoho e nyane mosebetsing \( f \) ka lebaka la phetoho ho \( x \).
Ho Bala Li-Derivative: Melao e Meng ea Motheo
Ho bala di-derivatives, ho na le melao e mmalwa ya motheo eo re ka e sebedisang:
1. Molao o sa Feleng
Haeba \( f(x) = c \), moo \( c \) e leng ntho e sa fetoheng, joale:
\[ f'(x) = 0 \]
Mohlala, haeba \( f(x) = 5 \), joale derivative ea \( f(x) \) ke 0.
2. Melao ea Maemo
Haeba \( f(x) = x^n \), moo \( n \) e leng palo e felletseng, joale:
\[ f'(x) = nx^{n-1} \]
Mohlala, haeba \( f(x) = x^3 \), joale:
\[ f'(x) = 3x^2 \]
3. Melao ea Lipalo
Haeba \( f(x) = g(x) + h(x) \), joale:
\[ f'(x) = g'(x) + h'(x) \]
Mohlala, haeba \( f(x) = x^2 + 3x \), joale:
\[ f'(x) = 2x + 3 \]
4. Melao ea Sehlahisoa
Haeba \( f(x) = g(x) \cdot h(x) \), joale:
\[ f'(x) = g'(x)h(x) + g(x)h'(x) \]
Mohlala, haeba \( f(x) = x^2 \cdot \sin(x) \), joale:
\[ f'(x) = 2x \cdot \sin(x) + x^2 \cdot \cos(x) \]
5. Molao oa Ketane
Haeba \( f(x) = g(h(x)) \), joale:
\[ f'(x) = g'(h(x)) \cdot h'(x) \]
Mohlala, haeba \( f(x) = \sin(x^2) \), joale:
\[ f'(x) = \cos(x^2) \cdot 2x \]
Tšebeliso ea Li-Derivative tsa Mosebetsi
Tšimoloho ea mosebetsi e na le lits'ebetso tse fapaneng bophelong ba 'nete le mafapheng a fapaneng a saense. Mehlala e meng ea lits'ebetso tsa eona ke ena:
1. Fisiks
Fisiks, di-derivative di sebediswa ho fumana lebelo le ho potlaka. A re re boemo ba ntho e le mosebetsi wa nako bo fanwa ke \( s(t) \). Ebe lebelo, \( v(t) \), ke derivative ya pele ya boemo:
\[ v(t) = s'(t) \]
Leha ho potlakisa, \( a(t) \), e le derivative ya bobedi ya boemo:
\[ a(t) = s”(t) = v'(t) \]
Mohlala, haeba \( s(t) = 4t^2 \), jwale lebelo ke \( v(t) = 8t \) mme ho potlaka ke \( a(t) = 8 \).
2. Moruo
Moruong, di-derivatives di sebediswa ho sekaseka ditjeo tse ka thoko le lekeno le ka thoko. A re re \(C(x) \) ke mosebetsi wa ditjeo tsohle bakeng sa ho hlahisa diyuniti tsa sehlahiswa \(x \). Ditjeo tse ka thoko, \(MC(x) \), ke derivative ya pele ya ditjeo tsohle:
\[ MC(x) = C'(x) \]
Ka ho tšoanang, haeba \( R(x) \) e le mosebetsi oa kakaretso ea lekeno ho tsoa ho rekiseng \( x \) liyuniti tsa sehlahisoa, joale lekeno le ka thoko, \( MR(x) \), ke derivative ea pele ea kakaretso ea lekeno:
\[ MR(x) = R'(x) \]
3. Baeloji
Thutong ea baeloji, li-derivative li sebelisoa ho etsa mohlala oa kholo ea baahi. A re re \( P(t) \) ke palo ea baahi ka nako \(t \), joale sekhahla sa kholo ea baahi ke derivative ea \( P(t) \):
\[ P'(t) \]
Sena se dumella ditsebi tsa baeloji ho utlwisisa hore na palo ya baahi e fetoha jwang ha nako e ntse e ya le hore na ke mabaka afe a e susumetsang.
4. Mokhoa
Boenjiniereng, di-derivative di sebediswa tlhahlobong le moralong wa ditsamaiso tsa taolo. Mohlala, moralong wa sistimi ya taolo ya PID (Proportional-Integral-Derivative), karolo ya derivative e fana ka karabelo e itshetlehileng ka sekgahla sa phetoho ya phoso. Sena se thusa ho ntlafatsa karabelo ya nakwana ya sistimi le ho fokotsa ho feteletsa.
Ho Rarolla Mathata: Mehlala e Sebetsang
Ho tebisa kutloisiso ea rona ea li-derivatives, ha re shebeng mehlala e meng ea lipotso.
Mohlala oa 1:
Fumana derivative ea \( f(x) = 5x^3 – 3x^2 + 6x – 2 \).
Tharollo:
Sebelisa melao ea exponent le kakaretso:
\[ f'(x) = 15x^2 – 6x + 6 \]
Mohlala oa 2:
Bala derivative ea \( f(x) = (3x^2 + 2x)(\sin(x)) \).
Tharollo:
Sebelisa melao ea sehlahisoa:
\[ f(x) = u(x)v(x) \]
moo \( u(x) = 3x^2 + 2x \) le \( v(x) = \sin(x) \)
\[ u'(x) = 6x + 2 \]
\[ v'(x) = \cos(x) \]
Kahoo:
\[ f'(x) = u'(x)v(x) + u(x)v'(x) = (6x + 2) \sin(x) + (3x^2 + 2x) \cos(x) \]
Qetello
Tšimoloho ea mosebetsi ke sesebelisoa se matla lipalo 'me se na le lits'ebetso tse ngata mafapheng a fapaneng. Ho utloisisa mokhoa oa ho bala metsoako le ho e sebelisa maemong a sebele ha ho bohlokoa feela khopolo-taba empa hape le ts'ebetsong ea letsatsi le letsatsi ea saense le boenjiniere. Ka melao e fapaneng ea motheo le mehlala e sebetsang, re ka tseba mohopolo oa metsoako le ho o sebelisa ho sekaseka liphetoho le ho bolela esale pele liphello maemong a fapaneng.
