Mehlala ea lipotso le puisano ea litšobotsi tsa mesebetsi e nkiloeng
Tšimoloho ea mosebetsi ke khopolo ea motheo ho calculus e thusang haholo bakeng sa ho sekaseka boitšoaro ba mesebetsi e itseng. Sehloohong sena, re tla tšohla mathata a 'maloa a mehlala le ho tšohla litšobotsi tsa motsoako oa mosebetsi.
Selelekela ho Li-Derivative tsa Mosebetsi
Setho sa mosebetsi \( f \) se hlahiswa e le \( f'(x) \). Setho sa pele sa mosebetsi se fana ka sekgahla sa phetoho ya mosebetsi mabapi le phetoho ya wona e ikemetseng. Lentswe le leng le sebediswang hangata ke phapang. Haeba \( y = f(x) \), jwale setho sa \( f \) mabapi le \( x \) ke:
\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) – f(x)}{h} \]
Matlotlo a Di-Derivative tsa Mosebetsi
Tse ling tsa litšobotsi tsa bohlokoa tsa derivative ea mosebetsi ke:
1. Ho lekana: Haeba \( f(x) \) le \( g(x) \) e le mesebetsi e ka kgetholohang, mme \( c \) e le ntho e sa fetoheng, jwale:
\[
\frac{d}{dx} [cf(x) + g(x)] = c f'(x) + g'(x)
\]
2. Molao oa Ketane: Bakeng sa mosebetsi o kopaneng \( g(f(x)) \):
\[
\frac{d}{dx} g(f(x)) = g'(f(x)) \cdot f'(x)
\]
3. Sehlahisoa: Bakeng sa mesebetsi \( u(x) \) le \( v(x) \):
\[
\frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]
4. Quotient: Bakeng sa mesebetsi \( u(x) \) le \( v(x) \) moo \( v(x) \neq 0 \):
\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) – u(x)v'(x)}{(v(x))^2}
\]
Lipotso tsa Mehlala le Puisano
Mohlala oa 1: Ho Fumana Tšimoloho ea Mosebetsi o Bonolo
A re re \( f(x) = 3x^2 + 5x – 4 \). Fumana derivative ya mosebetsi.
Tharollo:
Re tla sebelisa melao ea motheo ea ho khetholla.
\[
f(x) = 3x^2 + 5x – 4
\]
Ntho ea pele e nkiloeng:
\[
f'(x) = \frac{d}{dx} (3x^2) + \frac{d}{dx} (5x) – \frac{d}{dx} (4)
\]
Ho bala derivative ka 'ngoe:
\[
\frac{d}{dx} (3x^2) = 6x
\]
\[
\frac{d}{dx} (5x) = 5
\]
\[
\frac{d}{dx} (4) = 0
\]
E le hore:
\[
f'(x) = 6x + 5
\]
Mohlala oa 2: Ho Sebelisa Molao oa Ketane
Ha ho fanoe ka mosebetsi \( y = (2x^3 – x^2 + 1)^5 \). Fumana derivative ea mosebetsi.
Tharollo:
Sebelisa molao oa ketane. A re re \( u = 2x^3 – x^2 + 1 \), ebe ts'ebetso e ka ngoloa bocha e le \( y = u^5 \).
Taba ea pele, fumana derivative ea \( y \) mabapi le \( u \):
\[
\frac{dy}{du} = 5u^4
\]
Ka mor'a moo, fumana derivative ea \( u \) mabapi le \( x \):
\[
u = 2x^3 – x^2 + 1
\]
\[
\frac{du}{dx} = 6x^2 – 2x
\]
Kopanya di-derivatives tse pedi le molao wa ketane:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 5u^4 \cdot (6x^2 – 2x)
\]
Kenya sebaka hape \( u = 2x^3 – x^2 + 1 \):
\[
\frac{dy}{dx} = 5(2x^3 – x^2 + 1)^4 \cdot (6x^2 – 2x)
\]
Mohlala oa 3: Ho Sebelisa Melao ea Sehlahisoa
Fanoeng \( f(x) = x^2 e^x \). Fumana derivative ea mosebetsi.
Tharollo:
Sebelisa molao oa sehlahisoa, ke hore, haeba \( u(x) = x^2 \) le \( v(x) = e^x \), joale:
\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\]
Taba ea pele, bala li-derivative tsa \( u(x) \) le \( v(x) \):
\[
u(x) = x^2 \e bolela u'(x) = 2x
\]
\[
v(x) = e^x \ho bolela v'(x) = e^x
\]
Ka ho sebelisa melao ea sehlahisoa:
\[
f'(x) = 2x \cdot e^x + x^2 \cdot e^x = e^x (2x + x^2)
\]
Mohlala oa 4: Ho Sebelisa Molao oa Quotient
E fanoe \( f(x) = \frac{x^2 + 1}{x + 2} \). Fumana derivative ea mosebetsi.
Tharollo:
Sebelisa molao oa quotient, e leng haeba \( u(x) = x^2 + 1 \) le \( v(x) = x + 2 \), joale:
\[
f'(x) = \frac{u'(x)v(x) – u(x)v'(x)}{[v(x)]^2}
\]
Taba ea pele, bala li-derivative tsa \( u(x) \) le \( v(x) \):
\[
u(x) = x^2 + 1 \e bolela u'(x) = 2x
\]
\[
v(x) = x + 2 \e bolela v'(x) = 1
\]
Ka ho sebelisa molao oa quotient:
\[
f'(x) = \frac{2x(x + 2) – (x^2 + 1)(1)}{(x + 2)^2}
\]
\[
f'(x) = \frac{2x^2 + 4x – x^2 – 1}{(x + 2)^2}
\]
\[
f'(x) = \frac{x^2 + 4x – 1}{(x + 2)^2}
\]
Qetello
Ka hara dipalo, ho utlwisisa mohopolo wa motheo wa di-derivative le thepa ya tsona ho bohlokwa bakeng sa ho rarolla mathata a fapaneng a dipalo. Sengoloa sena se akaretsa mekgwa e mmalwa ya ho fumana mesebetsi ka ho bontsha tshebediso ya melao ya motheo e kang ho otloloha, diketane, dihlahiswa le di-quotients ka mehlala e mmalwa le dipuisano tse qaqileng. Ka ho utlwisisa le ho sebedisa di-derivatives kgafetsa, re ka ba le bokgoni bo eketsehileng ba ho sekaseka diphetoho mesebetsing maemong a fapaneng.
