Mehlala ea Lipotso le Puisano ea Khopolo-taba ea Li-Derivative tsa Mosebetsi
Tšimoloho ea mosebetsi ke khopolo ea motheo thutong ea lipalo e nang le lits'ebetso tse pharaletseng mafapheng a fapaneng, joalo ka fisiks, moruo le boenjiniere. Sehlooho sena se tla akaretsa mehlala e 'maloa ea mathata le ho buisana ka khopolo ea nts'etsopele ea mosebetsi ho fana ka kutloisiso e tebileng ea sehlooho sena.
Tlhaloso ea Motheo ea Li-Derivatives
Pele re kena lipotsong tsa mohlala, ke mohopolo o motle ho hlahloba ka bokhutšoanyane tlhaloso le metheo ea li-derivatives. Derivative ea mosebetsi \( f(x) \) ntlheng \( x = a \) ke:
\[ f'(a) = \lim_{{h \to 0}} \frac{f(a+h) – f(a)}{h} \]
Mosebetsi \( f'(x) \) o bitsoa mosebetsi o tswang ho \( f(x) \).
Mohlala oa Potso ea 1: Li-Derivative tsa Motheo tsa Polynomial
Potso:
Fumana derivative ea pele ea ts'ebetso \( f(x) = 3x^3 – 5x^2 + 2x – 7 \).
Puisano:
Sebelisa molao oa motheo oa derivative \( \frac{d}{dx} x^n = nx^{n-1} \).
1. Bakeng sa \( 3x^3 \):
\[ \frac{d}{dx}(3x^3) = 3 \cdot 3x^{3-1} = 9x^2 \]
2. Bakeng sa \( -5x^2 \):
\[ \frac{d}{dx}(-5x^2) = -5 \cdot 2x^{2-1} = -10x \]
3. Bakeng sa \( 2x \):
\[ \frac{d}{dx}(2x) = 2 \]
4. Bakeng sa \( -7 \):
\[ \frac{d}{dx}(-7) = 0 \]
Ka hona:
\[ f'(x) = 9x^2 – 10x + 2 \]
Mohlala oa Potso ea 2: Litholoana tsa Mesebetsi ea Trigonometric
Potso:
Fumana derivative ea pele ea ts'ebetso \( g(x) = \sin(x) \cdot \cos(x) \).
Puisano:
Sebelisa molao oa sehlahisoa \( \frac{d}{dx} [u(x) \cdot v(x)] = u'(x)v(x) + u(x)v'(x) \) le \( u(x) = \sin(x) \) le \( v(x) = \cos(x) \).
1. Setho sa \( \sin(x) \) ke \( \cos(x) \), kahoo \( u'(x) = \cos(x) \).
2. Setho sa \( \cos(x) \) ke \( -\sin(x) \), kahoo \( v'(x) = -\sin(x) \).
Phetolo \( u'(x) \) le \( v'(x) \):
\[ g'(x) = \cos(x) \cdot \cos(x) + \sin(x) \cdot (-\sin(x)) \]
\[ g'(x) = \cos^2(x) – \sin^2(x) \]
Sephetho sa ho qetela:
\[ g'(x) = \cos^2(x) – \sin^2(x) \]
Mohlala oa 3: Tšimoloho ea Mosebetsi oa Exponential
Potso:
Fumana derivative ea pele ea ts'ebetso \( h(x) = e^{2x} \).
Puisano:
Sebelisa molao oa derivative ea mosebetsi oa exponential \( \frac{d}{dx} e^{kx} = ke^{kx} \) le \( k = 2 \).
\[ h'(x) = \frac{d}{dx} e^{2x} \]
\[ h'(x) = 2 \cdot e^{2x} \]
Sephetho sa ho qetela:
\[ h'(x) = 2e^{2x} \]
Mohlala oa Potso ea 4: Tšimoloho ea Mosebetsi oa Logarithmic
Potso:
Fumana derivative ea pele ea ts'ebetso \( p(x) = \ln(3x + 1) \).
Puisano:
Sebelisa molao oa derivative ea mosebetsi oa logarithmic \( \frac{d}{dx} \ln(u) = \frac{1}{u} \cdot u' \) le \( u(x) = 3x + 1 \).
1. Fumana derivative ea ka hare \( u(x) = 3x + 1 \):
\[ u'(x) = 3 \]
2. Sebelisa molao oa derivative oa logarithmic:
\[ p'(x) = \frac{1}{3x + 1} \cdot 3 \]
Sephetho sa ho qetela:
\[ p'(x) = \frac{3}{3x + 1} \]
Mohlala Potso ea 5: Tšebeliso ea Li-derivatives - Boholo le Bonyane
Potso:
Fumana boleng bo phahameng ka ho fetisisa le bo bonyane ba mosebetsi \( q(x) = -2x^3 + 3x^2 + 12x – 5 \) karolong ya ho qetela \( x \in [-2, 2] \).
Puisano:
1. Fumana derivative ea pele ea \( q(x) \):
\[ q'(x) = \frac{d}{dx}(-2x^3 + 3x^2 + 12x – 5) \]
\[q'(x) = -6x^2 + 6x + 12 \]
2. Fumana lintlha tse sa sisinyeheng ka ho rarolla \( q'(x) = 0 \):
\[ -6x^2 + 6x + 12 = 0 \]
\[ -6(x^2 – x – 2) = 0 \]
\[ x^2 – x – 2 = 0 \]
\[ (x-2)(x+1) = 0 \]
Lintlha tse sa sisinyeheng ke \( x = 2 \) le \( x = -1 \).
3. Lekola \( q(x) \) dintlheng tsa bohlokwa le meeding ya karohano:
\[ q(-2) = -2(-2)^3 + 3(-2)^2 + 12(-2) – 5 \]
\[ = 16 + 12 – 24 – 5 \]
\[ = -1 \]
\[q(2) = -2(2)^3 + 3(2)^2 + 12(2) – 5 \]
\[ = -16 + 12 + 24 – 5 \]
\[ = 15 \]
\[ q(-1) = -2(-1)^3 + 3(-1)^2 + 12(-1) – 5 \]
\[ = 2 + 3 – 12 – 5 \]
\[ = -12 \]
4. Tlhahlobo ea liphetho:
– Boleng bo phahameng ka ho fetisisa bo hlaha ho \( x = 2 \) ka \( q(2) = 15 \).
– Boleng bo tlase bo hlaha ho \( x = -1 \) ka \( q(-1) = -12 \).
Ho koala
Kutloisiso e batsi ea mohopolo oa derivative ea mosebetsi ke ea bohlokoa mafapheng a fapaneng a saense. Re tšepa hore mehlala ea mathata le lipuisano li tla thusa ho tebisa kutloisiso ea hau ea mohopolo. Ha e le hantle, hangata re hloka ho kopanya melao le likhopolo-taba tse fapaneng ho rarolla mathata a rarahaneng haholoanyane. Thuto e monate!
