Lipotso tsa Mehlala le Puisano ea Molao oa Ketane ho Li-Derivatives
Molao oa ketane ke e 'ngoe ea likhopolo tsa motheo ka ho fetisisa ho differential calculus, e sebelisetsoang ho bala derivative ea mosebetsi o entsoeng ka mesebetsi e 'meli kapa ho feta. Sehloohong sena, re tla tšohla mohopolo oa motheo oa molao oa ketane, mokhoa oa ho o sebelisa, le mehlala ea tšebeliso ea oona mathateng a derivative a atisang ho hlaha sekolong se phahameng le kolecheng.
1. Selelekela sa Molao oa Ketane
Pele re kena bothateng ba mohlala, ha re utloisiseng pele hore na molao oa ketane ke eng. Molao oa ketane o bolela hore haeba re na le mesebetsi e 'meli e ka khetholloang \( f \) le \( g \), 'me re batla ho fumana derivative ea sebopeho sa mesebetsi \( h = f(g(x)) \), joale derivative ea \( h \) ke:
\[ h'(x) = f'(g(x)) \cdot g'(x) \]
Ka mantsoe a bonolo, re bala derivative ea mosebetsi oa kantle ho g(x), ebe re atisa sephetho ka derivative ea mosebetsi oa ka hare \( g(x) \).
2. Ho utloisisa Mosebetsi oa Sebopeho
Pele re kena mathateng a mohlala, ho bohlokoa ho utloisisa mesebetsi ea ho kopanya. Mosebetsi oa ho kopanya ke mosebetsi o fumanoang ka ho kenya mosebetsi o mong ho o mong. Mohlala, haeba re na le \( f(x) = \sin(x) \) le \( g(x) = x^2 \), joale sebopeho sa mesebetsi ena e 'meli e tla ba \( h(x) = f(g(x)) = \sin(x^2) \).
Mesebelong ea ho qapa, hangata re nahana ka \( g(x) \) e le "mosebetsi oa ka hare" le \( f(x) \) e le "mosebetsi oa kantle". Mohlaleng ona, mosebetsi oa ka hare ke \( x^2 \) 'me mosebetsi oa kantle ke sine.
3. Lipotso tsa Mehlala le Puisano
A re shebeng mehlala e meng ea mathata a sebelisang molao oa ketane ho a rarolla.
Mohlala oa 1:
Ha ho fanoe ka mosebetsi \( y = \cos(3x^2) \), fumana derivative ea pele ea y mabapi le x.
Puisano:
Taba ea pele, re khetholla mesebetsi ea ka hare le ea kantle. Mona, mosebetsi oa ka hare ke \( g(x) = 3x^2 \) 'me mosebetsi oa ka ntle ke \( f(g) = \cos(g) \).
Rea tseba:
1. \( g'(x) = 6x \)
2. \( f'(g) = -\sin(g) \)
Ho ya ka molao wa ketane, re fumana:
\[ y' = f'(g(x)) \cdot g'(x) = -\sin(3x^2) \cdot 6x \]
Kahoo, tlhahiso ea \( y = \cos(3x^2) \) ke:
\[ y' = -6x \sin(3x^2) \]
Mohlala oa 2:
Fumana derivative ea pele ea \( h(x) = e^{5x^3 + 2x} \).
Puisano:
Mona mosebetsi wa ka hare ke \( g(x) = 5x^3 + 2x \) mme mosebetsi wa ka ntle ke \( f(g) = e^g \).
Rea tseba:
1. \( g'(x) = 15x^2 + 2 \)
2. \( f'(g) = e^g \)
Ho ya ka molao wa ketane, re fumana:
\[ h'(x) = f'(g(x)) \cdot g'(x) = e^{5x^3 + 2x} \cdot (15x^2 + 2) \]
Kahoo, derivative ea \( h(x) = e^{5x^3 + 2x} \) ke:
\[ h'(x) = (15x^2 + 2) e^{5x^3 + 2x} \]
Mohlala oa 3:
Fumana derivative ea pele ea \( y = \ln(4x^2 - 5) \).
Puisano:
Mosebetsi wa ka hare ke \( g(x) = 4x^2 – 5 \) mme mosebetsi wa ka ntle ke \( f(g) = \ln(g) \).
Rea tseba:
1. \( g'(x) = 8x \)
2. \( f'(g) = \frac{1}{g} \)
Ho ya ka molao wa ketane, re fumana:
\[ y' = f'(g(x)) \cdot g'(x) = \frac{1}{4x^2 – 5} \cdot 8x \]
Kahoo, tlhahiso ea \( y = \ln(4x^2 – 5) \) ke:
\[ y' = \frac{8x}{4x^2 – 5} \]
Mohlala oa 4:
Ha ho fanoe ka mosebetsi \( y = (3x^2 + 2x + 1)^4 \), fumana derivative ea eona.
Puisano:
Mosebetsi wa ka hare ke \( g(x) = 3x^2 + 2x + 1 \) mme mosebetsi wa ka ntle ke \( f(g) = g^4 \).
Rea tseba:
1. \( g'(x) = 6x + 2 \)
2. \( f'(g) = 4g^3 \)
Ho ya ka molao wa ketane, re fumana:
\[ y' = f'(g(x)) \cdot g'(x) = 4(3x^2 + 2x + 1)^3 \cdot (6x + 2) \]
Kahoo, derivative ea \( y = (3x^2 + 2x + 1)^4 \) ke:
\[ y' = 4(3x^2 + 2x + 1)^3 (6x + 2) \]
4. Linyeoe tse Ikhethang le Ntlafatso ea Melao ea Ketane
Ka linako tse ling, molao oa ketane ha o emise ho sebopeho sa mesebetsi e 'meli feela. Ho na le linako tseo mosebetsi e leng sebopeho sa mesebetsi e fetang e 'meli, mohlala: \( h(x) = f(g(k(x))) \).
Tabeng ea mesebetsi e meraro, molao oa ketane o ka sebelisoa ka mekhahlelo:
\[ h'(x) = f'(g(k(x))) \cdot g'(k(x)) \cdot k'(x) \]
Re ka bona hore lera ka leng, re bala di-derivative tsa dikarolo tse ka ntle pele re fetela ho di-derivative tsa dikarolo tse ka hare.
Mohlala oa 5:
Ha ho fanoe ka \( y = \sqrt{\ln(2x^2 + 1)} \), fumana derivative ea eona.
Puisano:
Mosebetsi o ka hare-hare ke \( k = 2x^2 + 1 \), bohareng: \( g = \ln(k) \) le kantle: \( f = \sqrt{g} \).
Rea tseba:
1. \( k'(x) = 4x \)
2. \( g'(k) = \frac{1}{k} \)
3. \( f'(g) = \frac{1}{2\sqrt{g}} \)
Ha re sebeliseng molao oa ketane ka mekhahlelo:
\[ y' = f'(g(k(x))) \cdot g'(k(x)) \cdot k'(x) = \frac{1}{2\sqrt{\ln(2x^2 + 1)}} \cdot \frac{1}{2x^2 + 1} \cdot 4x \]
Kahoo tlhahiso ea \( y = \sqrt{\ln(2x^2 + 1)} \) ke:
\[ y' = \frac{4x}{2(2x^2 + 1)\sqrt{\ln(2x^2 + 1)}} \]
5. Kesimpulan
Molao oa ketane o bapala karolo ea bohlokoa ho differential calculus, haholo-holo ha ho sebetsanoa le ho hlophisoa ha mesebetsi. Ho utloisisa le ho tseba molao oa ketane ho fana ka motheo o tiileng oa ho sebetsana le mathata a rarahaneng haholoanyane ho calculus. Sengoloa sena se buile ka mehlala e 'maloa ea bohlokoa ho fana ka kutloisiso e tiileng ea ts'ebeliso ea molao oa ketane ho li-derivatives. Re tšepa hore puisano ena e thusa baithuti 'me e ka sebelisoa maemong a fapaneng a thata a lipalo.
