Mehlala ea Lipotso Tse Buisanang ka Li-Derivative tsa Mosebetsi oa ho Ngola
Tšimoloho ea mosebetsi ke mohopolo oa motheo ho calculus, o sebelisoang khafetsa mafapheng a fapaneng a mahlale, joalo ka fisiks, moruo, baeloji le boenjiniere. Tšimoloho ea mosebetsi e lekanya hore na boleng ba oona bo fetoha kapele hakae mabapi le liphetoho tse feto-fetohang tsa oona tse ikemetseng. Sehloohong sena, re tla tšohla mehlala e 'maloa ea mathata a amanang le ho ngola ts'ebetso e tsoang ho mosebetsi, e felletseng ka litlhaloso.
Mohlala oa Potso ea 1: Tšimoloho ea Mesebetsi e Bonolo
Potso: Fumana derivative ea pele ea ts'ebetso \( f(x) = 3x^2 + 5x + 7 \).
Puisano:
Ho fumana derivative ea pele ea mosebetsi \( f(x) \), re sebelisa melao ea motheo ea phapang, e leng:
\[
\frac{d}{dx}(ax^n) = anx^{n-1}
\]
Kahoo, re ka bala derivative ea lentsoe ka leng mosebetsing ka tsela e latelang:
\[
f'(x) = \frac{d}{dx}(3x^2) + \frac{d}{dx}(5x) + \frac{d}{dx}(7)
\]
\[
f'(x) = 3 \cdot 2x^{2-1} + 5 \cdot 1x^{1-1} + 0
\]
\[
f'(x) = 6x + 5
\]
Kahoo, derivative ea pele ea ts'ebetso \( f(x) = 3x^2 + 5x + 7 \) ke \( f'(x) = 6x + 5 \).
Mohlala oa Potso ea 2: Litholoana tsa Mesebetsi ea Trigonometric
Potso: Fumana derivative ea pele ea ts'ebetso \( g(x) = \sin(x) + \cos(x) \).
Puisano:
Re sebelisa melao ea motheo ea derivative bakeng sa mesebetsi ea trigonometric:
\[
\frac{d}{dx}(\sin(x)) = \cos(x)
\]
\[
\frac{d}{dx}(\cos(x)) = -\sin(x)
\]
Kahoo:
\[
g'(x) = \frac{d}{dx}(\sin(x)) + \frac{d}{dx}(\cos(x))
\]
\[
g'(x) = \cos(x) – \sin(x)
\]
Kahoo, derivative ea pele ea ts'ebetso \( g(x) = \sin(x) + \cos(x) \) ke \( g'(x) = \cos(x) – \sin(x) \).
Mohlala oa Potso ea 3: Tšimoloho ea Mosebetsi oa ho Atisa
Potso: Fumana derivative ea pele ea ts'ebetso \( h(x) = x^2 \sin(x) \).
Puisano:
Bakeng sa mesebetsi e leng sehlahisoa sa mesebetsi e 'meli, re sebelisa molao oa ho atisa:
\[
\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
\]
A re re \( u(x) = x^2 \) le \( v(x) = \sin(x) \). Ebe:
\[
u'(x) = \frac{d}{dx}(x^2) = 2x
\]
\[
v'(x) = \frac{d}{dx}(\sin(x)) = \cos(x)
\]
Re sebelisa molao oa ho atisa, re ka ngola:
\[
h'(x) = [x^2]' \sin(x) + x^2 [\sin(x)]'
\]
\[
h'(x) = 2x \sin(x) + x^2 \cos(x)
\]
Kahoo, derivative ea pele ea ts'ebetso \( h(x) = x^2 \sin(x) \) ke \( h'(x) = 2x \sin(x) + x^2 \cos(x) \).
Mohlala Potso ea 4: Tšimoloho ea Mosebetsi oa Sebopeho
Potso: Fumana derivative ea pele ea ts'ebetso \( k(x) = \sin(x^2) \).
Puisano:
Bakeng sa mesebetsi e leng sebopeho sa mesebetsi e 'meli, re sebelisa molao oa ketane:
\[
\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)
\]
E re \( f(u) = \sin(u) \) le \( u = x^2 \). Ebe \( f'(u) = \cos(u) \) le \( g'(x) = \frac{d}{dx}(x^2) = 2x \).
Re sebelisa molao oa ketane, re ka ngola:
\[
k'(x) = \frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x
\]
Kahoo, derivative ea pele ea ts'ebetso \( k(x) = \sin(x^2) \) ke \( k'(x) = 2x \cos(x^2) \).
Mohlala oa Potso ea 5: Tšimoloho ea Mesebetsi e utloahalang
Bothata: Fumana derivative ea pele ea ts'ebetso \( m(x) = \frac{2x}{x^2 + 1} \).
Puisano:
Bakeng sa mesebetsi e leng quotient ea mesebetsi e 'meli, re sebelisa molao oa quotient:
\[
\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) – u(x)v'(x)}{[v(x)]^2}
\]
A re re \( u(x) = 2x \) le \( v(x) = x^2 + 1 \). Ebe:
\[
u'(x) = 2
\]
\[
v'(x) = \frac{d}{dx}(x^2 + 1) = 2x
\]
Re sebelisa molao oa quotient, re ka ngola:
\[
m'(x) = \frac{[2x]'(x^2 + 1) – 2x[x^2 + 1]'}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2(x^2 + 1) – 2x \cdot 2x}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2x^2 + 2 – 4x^2}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{-(2x^2 – 2)}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2 – 2x^2}{(x^2 + 1)^2}
\]
Kahoo, derivative ea pele ea ts'ebetso \( m(x) = \frac{2x}{x^2 + 1} \) ke \( m'(x) = \frac{2 – 2x^2}{(x^2 + 1)^2} \).
Qetello
Sehloohong sena, re buisane ka mehlala e 'maloa ea mathata a amanang le li-derivative tsa mesebetsi, ho tloha mesebetsing e bonolo, mesebetsi ea trigonometric, katiso, sebopeho, le mesebetsi e utloahalang. Mohlala ka mong o bontša tšebeliso e nepahetseng ea melao ea derivative, joalo ka molao oa motheo, molao oa ketane, molao oa katiso, le molao oa quotient. Ho utloisisa mokhoa oa ho sebelisa melao ena ho bohlokoa bakeng sa ho rarolla mathata a rarahaneng a calculus mafapheng a fapaneng. Boitlhakiso le koetliso e pheta-phetoang li tla thusa ho matlafatsa kutloisiso ea hau le bokhoni ba ho khetholla mesebetsi.
