Li-derivatives tsa Mesebetsi ea Algebraic: Tataiso e Felletseng
Tšimoloho ea mosebetsi ke khopolo ea motheo thutong ea lipalo le lipalo ka kakaretso. Khopolo ena ha e sebetse feela khopolo-taba empa e boetse e na le lits'ebetso tse sebetsang mafapheng a fapaneng, ho kenyeletsoa fisiks, boenjiniere, moruo le mahlale a khomphutha. Sehlooho sena se tla tšohla tlhaloso ea algebraic ea mosebetsi, ho tloha tlhalosong ea eona ea motheo ho isa lits'ebetsong tsa eona tse rarahaneng haholoanyane.
Tlhaloso ea Li-derivatives
Ho lipalo, derivative ea mosebetsi e emela sekhahla sa phetoho ea mosebetsi mabapi le phetoho ea oona e ikemetseng. Ka kutloisiso, derivative e ka nahanoa e le leralla la mola o tangent ho ea ho kobeho ntlheng e fanoeng. Haeba \( y = f(x) \), joale derivative ea mosebetsi e hlalosoa e le \( f'(x) \) kapa \( \frac{dy}{dx} \).
Mokhoa oa ho Fokotsa Moeli
Tlhaloso e hlophisitsoeng ea derivative e sebelisa mohopolo oa moeli. Haeba \( f(x) \) e le mosebetsi o tsoelang pele, joale derivative ea pele ea mosebetsi e hlalosoa e le:
\[
f'(x) = \lim_{{h \to 0}} \frac{f(x+h) – f(x)}{h}
\]
Mona, \( h \) ke phetoho e nyane ho \( x \). Moeli ona, haeba o le teng, o fana ka sekgahla se setle ka ho fetisisa sa phetoho kapa moedi wa \( f(x) \) ntlheng \( x \).
Melao ea Motheo ea Phapang
1. Molao o sa Feleng:
Haeba \( c \) e le ntho e sa fetoheng le \( f(x) = c \), joale:
\[
f'(x) = 0
\]
2. Melao ea Maemo:
Haeba \( f(x) = x^n \) bakeng sa nomoro efe kapa efe ya nnete \( n \), joale:
\[
f'(x) = nx^{n-1}
\]
3. Molao o sa Fetoheng habeli:
Haeba \( f(x) = cg(x) \) bakeng sa mosebetsi ofe kapa ofe \( g(x) \) le kamehla \( c \), joale:
\[
(cf(x))' = c f'(x)
\]
4. Melao ea ho Eketsa:
Haeba \( f(x) \) le \( g(x) \) e le mesebetsi e 'meli e ka kgetholohang, joale:
\[
(f(x) + g(x))' = f'(x) + g'(x)
\]
5. Melao ea ho Atisa:
Haeba \( f(x) \) le \( g(x) \) e le mesebetsi e 'meli e ka kgetholohang, joale:
\[
(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)
\]
6. Melao ea Karohano:
Bakeng sa mesebetsi e 'meli \( f(x) \) le \( g(x) \) e ka khetholloang ka \( g(x) \neq 0 \), ebe:
\[
\left( \frac{f(x)}{g(x)} \right)' = \frac{f'(x)g(x) – f(x)g'(x)}{g(x)^2}
\]
7. Molao oa Ketane:
Haeba \( y = f(u) \) le \( u = g(x) \), joale:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
\]
Mehlala ea Ts'ebeliso
Mohlala oa 1: A re re \( f(x) = 4x^3 – 2x + 7 \). Ebe derivative ea \( f(x) \) e ka baloa ka tsela e latelang:
\[
f'(x) = 12x^2 – 2
\]
Mona, re sebelisa molao oa matla le molao oa kamehla o habeli.
_Mohlala oa 2:_ Nahana ka \( g(x) = (2x^2 – 3x)(x^3 + 1) \). Ho fumana \( g'(x) \), re sebelisa molao oa ho atisa:
\[
g'(x) = (2x^2 – 3x)'(x^3 + 1) + (2x^2 – 3x)(x^3 + 1)'
\]
\[
= (4x – 3)(x^3 + 1) + (2x^2 – 3x)(3x^2)
\]
\[
= 4x(x^3 + 1) – 3(x^3 + 1) + 6x^4 – 9x^3
\]
\[
= 4x^4 + 4x – 3x^3 – 3 + 6x^4 – 9x^3
\]
\[
= 10x^4 – 12x^3 + 4x – 3
\]
Litšebeliso tsa Bophelo ba Sebele ba Derivatives
1. Fisiks:
Fisiks hangata e sebedisa di-derivative ho utlwisisa mehopolo ya lebelo le ho potlaka. Mohlala, haeba \( s(t) \) ke boemo ba ntho e le mosebetsi wa nako \(t \), jwale lebelo \(v(t) \) ke derivative ya pele ya boemo \( s(t) \), mme ho potlaka \( a(t) \) ke derivative ya lebelo.
2. Moruo:
Moruong, di-derivatives di sebediswa ho fumana sekgahla sa phetoho se ka thoko. Mehlala ya ditshebediso e kenyeletsa ditjeo tse ka thoko, tse hlalosang kamoo ditjeo tsohle di fetohang ka tlhahiso ya yuniti e le nngwe e eketsehileng.
3. Mokhoa:
Boenjiniereng, di-derivative di sebediswa bakeng sa tlhahlobo ya botsitso le taolo ya sistimi. Mohlala, ho mekhaniki ya sebopeho, di-derivative di sebediswa ho fumana kgatello le mathata dinthong.
4. Likerafo le Likhutlo:
Di-derivative di boetse di sebediswa ho fumana dintlha tse phahameng le tse tlase hodima lekoko la mosebetsi, e leng ntho ya bohlokwa ntlafatsong.
Qetello
Ho tseba mohopolo oa derivative ea mesebetsi ea algebra ho bohlokoa bakeng sa ho utloisisa liketsahalo tse fapaneng tsa lipalo le ts'ebeliso ea tsona ea bophelo ba sebele. Ka ho sebelisa melao ea motheo ea phapang, re ka fumana habonolo derivatives ea mesebetsi e fapaneng 'me ra e sebelisa ho rarolla mathata a lefats'e la nnete masimong a fapaneng. Re tšepa hore sengoloa sena se fana ka kutloisiso e felletseng ea derivatives ea mesebetsi ea algebra.
Referensi
Ho tebisa tsebo ea hau ea li-derivatives, re khothaletsa haholo ho bala buka ea buka ea lipalo e kang "Calculus" ea James Stewart kapa "Advanced Calculus" ea Michael Spivak. Ho feta moo, mehloli e fapaneng ea inthanete le lithuto tsa video li ka thusa haholo.
