Mohlala oa potso ea puisano mabapi le derivative ea mosebetsi oa algebraic
Derivative ho calculus ke mohopolo wa motheo o sebediswang ho hlalosa kamoo mosebetsi o fetohang kateng, kapa moedi wa mosebetsi ntlheng e itseng. Di-derivative di na le thuso mafapheng a fapaneng a kang fisiks, moruo le boenjiniere hobane di fana ka tlhahisoleseding mabapi le sekgahla sa phetoho. Sehloohong sena, re tla buisana ka mehlala e mmalwa ya di-derivative tsa mesebetsi ya algebra le mokhoa wa ho di rarolla.
Mohlala oa 1: Tšimoloho ea Mosebetsi oa Polynomial
Potso: Ha ho fanoe ka mosebetsi \( f(x) = 3x^3 – 5x^2 + 2x – 7 \). Fumana derivative ya mosebetsi!
Tharollo:
Re sebelisa molao oa motheo oa li-derivative bakeng sa mesebetsi ea polynomial, e leng \(\frac{d}{dx} x^n = nx^{n-1} \), re tla bala derivative ea lentsoe ka leng la mosebetsi ka bonngoe.
\[
\begin{align}
f(x) &= 3x^3 – 5x^2 + 2x – 7 \\
f'(x) &= \frac{d}{dx}(3x^3) – \frac{d}{dx}(5x^2) + \frac{d}{dx}(2x) – \frac{d}{dx}(7) \\
f'(x) &= 3 \cdot 3x^{3-1} – 5 \cdot 2x^{2-1} + 2 \cdot 1x^{1-1} – 0 \\
f'(x) &= 9x^2 – 10x + 2.
\end{align}
\]
Kahoo, derivative ea \( f(x) = 3x^3 – 5x^2 + 2x – 7 \) ke \( f'(x) = 9x^2 – 10x + 2 \).
Mohlala oa 2: Tšimoloho ea Mosebetsi ka Li-Exponents tsa Likaroloana
Potso: Fumana derivative ea mosebetsi \( g(x) = x^{3/2} + x^{1/2} \).
Tharollo:
Ho sebelisoa molao o tšoanang oa ho tsoa ho oona, ke hore \(\frac{d}{dx} x^n = nx^{n-1} \):
\[
\begin{align}
g(x) &= x^{3/2} + x^{1/2} \\
g'(x) &= \frac{d}{dx}(x^{3/2}) + \frac{d}{dx}(x^{1/2}) \\
g'(x) &= \frac{3}{2}x^{(3/2)-1} + \frac{1}{2}x^{(1/2)-1} \\
g'(x) &= \frac{3}{2}x^{1/2} + \frac{1}{2}x^{-1/2}.
\end{align}
\]
Kahoo, derivative ea \( g(x) = x^{3/2} + x^{1/2} \) ke \( g'(x) = \frac{3}{2}x^{1/2} + \frac{1}{2}x^{-1/2} \).
Mohlala oa 3: Litholoana tsa Mesebetsi ea Exponential le Trigonometric
Potso: Fumana derivative ea mosebetsi \( h(x) = e^x \cdot \sin(x) \).
Tharollo:
Ho rarolla derivative ena, re hloka Molao oa Sehlahisoa, o reng \((uv)' = u'v + uv'\). A re re \( u(x) = e^x \) le \( v(x) = \sin(x) \), ebe:
\[
\begin{align}
u'(x) &= e^x, & \text{hobane derivative ea } e^x \text{ ke } e^x \\
v'(x) &= \cos(x), & \text{hobane se tswang ho } \sin(x) \text{ ke } \cos(x).
\end{align}
\]
Ho sebelisa molao o nkiloeng bakeng sa lihlahisoa:
\[
\begin{align}
h'(x) &= (e^x \cdot \sin(x))' \\
&= e^x \cdot (\sin(x))' + \sin(x) \cdot (e^x)' \\
&= e^x \cdot \cos(x) + \sin(x) \cdot e^x \\
&= e^x (\cos(x) + \sin(x)).
\end{align}
\]
Kahoo, tlhahiso ea \( h(x) = e^x \sin(x) \) ke \( h'(x) = e^x (\cos(x) + \sin(x)) \).
Mohlala oa 4: Tšimoloho ea Mosebetsi o Sebelisang Molao oa Ketane
Potso: Fumana derivative ea mosebetsi \( k(x) = (3x^2 – x + 4)^5 \).
Tharollo:
Ho rarolla derivative ena, re hloka molao wa ketane, e leng \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\). A re re \( u(x) = 3x^2 – x + 4 \) le \( f(u) = u^5 \), ebe:
\[
\begin{align}
k(x) &= (3x^2 – x + 4)^5 \\
u(x) &= 3x^2 – x + 4, & \text{so} \\
k(x) &= f(u(x)) = u^5 \\
k'(x) &= 5u^4 \cdot u'(x) \\
u'(x) &= \frac{d}{dx}(3x^2 – x + 4) \\
&= 6x – 1.
\end{align}
\]
Ho sebelisa molao oa ketane:
\[
\begin{align}
k'(x) &= 5(3x^2 – x + 4)^4 \cdot (6x – 1) \\
&= 5(3x^2 – x + 4)^4 (6x – 1).
\end{align}
\]
Kahoo, derivative ea \( k(x) = (3x^2 – x + 4)^5 \) ke \( k'(x) = 5 (3x^2 – x + 4)^4 (6x – 1) \).
Mohlala oa 5: Tšimoloho ea Mosebetsi o nang le Boitsebiso ba Trigonometric
Potso: Fumana derivative ea mosebetsi \( m(x) = \sin(x) \cdot \cos(x) \).
Tharollo:
Re tla sebelisa molao oa derivative bakeng sa lihlahisoa. A re re \( u(x) = \sin(x) \) le \( v(x) = \cos(x) \), ebe:
\[
\begin{align}
u'(x) &= \cos(x), \\
v'(x) &= -\sebe(x).
\end{align}
\]
Ho sebelisa molao o nkiloeng bakeng sa lihlahisoa:
\[
\begin{align}
m'(x) &= (\sin(x) \cdot \cos(x))' \\
&= (\sin(x))' \cdot \cos(x) + \sin(x) \cdot (\cos(x))' \\
&= \cos(x) \cdot \cos(x) + \sin(x) \cdot (-\sin(x)) \\
&= \cos^2(x) – \sin^2(x).
\end{align}
\]
Ho sebedisa boitsebiso ba trigonometric \(\cos(2x) = \cos^2(x) – \sin^2(x)\):
\[
m'(x) = \cos(2x).
\]
Kahoo, tlhahiso ea \( m(x) = \sin(x) \cdot \cos(x) \) ke \( m'(x) = \cos(2x) \).
Qetello
Tšimoloho ea mosebetsi oa algebraic ke khopolo ea motheo ka har'a calculus e bohlokoa haholo ebile e na le thuso lits'ebetsong tse fapaneng. Melao e fapaneng ea ho tsoa, joalo ka molao oa motheo oa derivative, molao oa sehlahisoa, molao oa ketane, le melao ea li-derivative tsa trigonometric, kaofela li thusa ho bala li-derivative tsa mesebetsi e rarahaneng haholoanyane. Ka ho utloisisa mehlala e kaholimo le ho sebelisa mathata, re ka ntlafatsa kutloisiso ea rona le bokhoni ba ho nka li-derivative tsa mesebetsi ea algebraic.
