Mafunso a Zitsanzo ndi Kukambirana za Zochokera ku Ntchito
Chochokera ku deta ndi lingaliro lofunikira mu calculus lomwe limagwira ntchito yofunika kwambiri pakugwiritsa ntchito masamu, fizikisi, uinjiniya, ndi sayansi zina. M'nkhaniyi, tikambirana zitsanzo zingapo za zochokera ku deta ndi mayankho ake. Kumvetsetsa lingaliro la chochokera ku deta kudzakuthandizani kuligwiritsa ntchito mosavuta pamavuto osiyanasiyana.
Kumvetsetsa Koyambira kwa Zotumphukira
Chochokera ku ntchito chimafotokoza kuchuluka kwa kusintha kwa ntchitoyo poyerekeza ndi chosinthika chodziyimira pawokha. Mwachidziwitso, chochokera ku ntchito \( f(x) \) pa mfundo \( x \) ndi malo otsetsereka a mzere wozungulira kupita ku curve \( f \) pa mfundo \( x \). Zolemba zodziwika bwino zomwe zimagwiritsidwa ntchito pa chochokera ndi \( f'(x) \) kapena \( \frac{df}{dx} \).
Malamulo Oyambira a Zotumphukira
Kuti tithetse mavuto okhudzana ndi ma derivatives, tiyenera kudziwa malamulo oyambira a ma derivatives:
1. Chochokera Chokhazikika: Ngati \( c \) ndi chokhazikika, ndiye kuti chochokera cha \( c \) ndi zero.
\[
\frac{d}{dx}(c) = 0
\]
2. Chochokera ku Ntchito Yolunjika: Ngati \( f(x) = mx + b \), pomwe \( m \) ndi \( b \) ndi zosasinthika, ndiye kuti:
\[
f'(x) = m
\]
3. Lamulo la Mphamvu: Ngati \( f(x) = x^n \), pomwe \( n \) ndi nambala yeniyeni, ndiye kuti:
\[
f'(x) = nx^{n-1}
\]
4. Lamulo Lophatikiza: Ngati \( f(x) = g(x) + h(x) \), ndiye kuti:
\[
f'(x) = g'(x) + h'(x)
\]
5. Lamulo Lochulukitsa: Ngati \( f(x) = g(x) \cdot h(x) \), ndiye:
\[
f'(x) = g'(x)h(x) + g(x)h'(x)
\]
6. Lamulo la Kugawa: Ngati \( f(x) = \frac{g(x)}{h(x)} \), ndiye kuti:
\[
f'(x) = \frac{g'(x)h(x) – g(x)h'(x)}{h(x)^2}
\]
7. Lamulo la Unyolo: Ngati \( f(x) = g(h(x)) \), ndiye kuti:
\[
f'(x) = g'(h(x)) \cdot h'(x)
\]
Mafunso ndi Kukambirana Zitsanzo
Chitsanzo cha Funso 1
Funso: Dziwani zomwe zimachokera ku \( f(x) = 3x^2 + 2x + 1 \).
Kukambirana:
Kuti tidziwe zomwe zimachokera ku ntchitoyo, tigwiritsa ntchito lamulo la mphamvu ndi lamulo lokwanira.
\[
f(x) = 3x^2 + 2x + 1
\]
Zotumphukira zake ndi izi:
\[
f'(x) = \frac{d}{dx}(3x^2) + \frac{d}{dx}(2x) + \frac{d}{dx}(1)
\]
Kutengera ndi lamulo la udindo:
\[
\frac{d}{dx}(3x^2) = 3 \cdot 2x^{2-1} = 6x
\]
\[
\frac{d}{dx}(2x) = 2 \cdot 1x^{1-1} = 2
\]
\[
\frac{d}{dx}(1) = 0
\]
Kotero, chochokera ku ntchito \( f \) ndi:
\[
f'(x) = 6x + 2
\]
Chitsanzo cha Funso 2
Funso: Dziwani zomwe zimachokera ku ntchito \( g(x) = (2x^3 – x)(x^2 + 3) \).
Kukambirana:
Kuti tithetse vutoli, tigwiritsa ntchito lamulo lochulukitsa.
\[
g(x) = (2x^3 – x)(x^2 + 3)
\]
Kotero, chochokera pa \( g(x) \) ndi:
\[
g'(x) = (2x^3 – x)'(x^2 + 3) + (2x^3 – x)(x^2 + 3)'
\]
Choyamba, timapeza zomwe zimachokera ku ntchito iliyonse:
\[
(2x^3 – x)' = 6x^2 – 1
\]
\[
(x^2 + 3)' = 2x
\]
Kenako timasintha ndi fomula iyi:
\[
g'(x) = (6x^2 – 1)(x^2 + 3) + (2x^3 – x)(2x)
\]
Kenako, timagawa:
\[
g'(x) = 6x^2 \cdot x^2 + 6x^2 \cdot 3 – 1 \cdot x^2 – 1 \cdot 3 + 2x^3 \cdot 2x – x \cdot 2x
\]
\[
g'(x) = 6x^4 + 18x^2 – x^2 – 3 + 4x^4 – 2x^2
\]
Pomaliza, timapeza:
\[
g'(x) = 10x^4 + 15x^2 – 3
\]
Chitsanzo cha Funso 3
Funso: Pezani chochokera ku \( h(x) = \frac{x^2 + 1}{x – 1} \).
Kukambirana:
Kuti tithetse vutoli, tigwiritsa ntchito lamulo logawa.
\[
h(x) = \frac{x^2 + 1}{x – 1}
\]
Kotero, chochokera pa \( h(x) \) ndi:
\[
h'(x) = \frac{(x^2 + 1)'(x – 1) – (x^2 + 1)(x – 1)'}{(x – 1)^2}
\]
Choyamba, timapeza zomwe zimachokera ku ntchito iliyonse:
\[
(x^2 + 1)' = 2x
\]
\[
(x – 1)' = 1
\]
Kenako timasintha ndi fomula iyi:
\[
h'(x) = \frac{2x(x – 1) – (x^2 + 1)(1)}{(x – 1)^2}
\]
Kenako, timagawa:
\[
h'(x) = \frac{2x^2 – 2x – x^2 – 1}{(x – 1)^2}
\]
Kenako timasavuta:
\[
h'(x) = \frac{x^2 – 2x – 1}{(x – 1)^2}
\]
Mapeto
Chochokera ku ntchito ndi lingaliro lofunikira mu calculus lomwe limapereka chidziwitso chokhudza kusintha kwa mtengo wa ntchitoyo poyerekeza ndi chosinthika chake chodziyimira pawokha. Mwa kumvetsetsa malamulo oyambira a chochokera, monga chochokera ku ntchito zosasintha, zolunjika, lamulo la mphamvu, kuchuluka, kuchulukitsa, ndi kugawa, ndi lamulo la unyolo, titha kuthetsa mavuto osiyanasiyana ochokera ku ntchito.
Zitsanzo za mavuto omwe afotokozedwa pamwambapa ndi gawo loyamba labwino pomvetsetsa momwe mungagwiritsire ntchito lingaliro la zotumphukira. M'machitidwe, luso lowerengera zotumphukira lidzakulitsidwa kwambiri pogwira ntchito ndi mitundu yosiyanasiyana ya mavuto ndi kusiyanasiyana kwa ntchito. Tikukhulupirira kuti nkhaniyi yakhala yothandiza kumvetsetsa ndikudziwa bwino lingaliro la zotumphukira za ntchito.
