Mutemo weChain muZvinyorwa
Masvomhu ane basa rakakosha muzvikamu zvakasiyana-siyana zvehupenyu, kubva kusayenzi kusvika kuhupfumi. Imwe nyaya inokosha mukuverenga ndeyepfungwa yederivative. Derivatives inopa nzira yekunzwisisa kuti mabasa anochinja sei sezvo variables ichichinja. Imwe yepfungwa dzakakosha mukuverenga kwedifferential icheni, iyo inopa nzira yekuverenga derivatives yemabasa akabatanidzwa. Chinyorwa chino chichaongorora cheni rule zvakadzama, kubva pakutsanangurwa kwayo kusvika pakushandiswa kwayo muminda yakasiyana-siyana.
Tsananguro yeMutemo weChain
Mutemo wecheni ndeimwe yemitemo mikuru mu differential calculus inoshandiswa kuverenga derivative ye composition yemabasa maviri kana anopfuura. Pamutemo, kana tine mabasa maviri \( f(x) \) uye \( g(x) \), uye tichida kuwana derivative yecomposite function \( h(x) = f(g(x)) \), ipapo mutemo wecheni unoti:
\[ h'(x) = f'(g(x)) \cdot g'(x) \]
Nemamwe mashoko, kuti tiverenge derivative ye \( h(x) \), tinofanira kuwanza derivative ye \( f \) yakaongororwa pa \( g(x) \) ne derivative ye \( g \).
Kunzwisisa kweJomethri kweMutemo weCheni
Kuti unzwisise mutemo wecheni nenzira yekunzwisisa, fungidzira kuti tiri kufamba mumugwagwa unomonereka. Kumhanya kwatinoita tichifambira mberi mumugwagwa uyu (kureva, zvinobva panzvimbo yedu maererano nenguva) kunoenderana nezvinhu zviviri: kumhanya kwedu munzira yemugwagwa, uye kutsetseka kwemugwagwa panzvimbo yakatarwa. Saizvozvowo, mumamiriro ezvinhu emutemo wecheni, shanduko mubasa rejoini \( h(x) \) inokonzerwa nezvinhu zviviri: kuti \( f \) shanduko sei maererano ne \( g \), uye kuti \( g \) shanduko sei maererano ne \( x \).
Muenzaniso Uri Nyore
Ngatitarisei muenzaniso uri nyore patinoshandisa mutemo wecheni kuverenga derivative yecompound function.
Ngatitii \( f(x) = \sin(x) \) uye \( g(x) = x^2 \). Zvadaro, tinoda kuwana chinobva pa \( h(x) = \sin(x^2) \).
Heano matanho:
1. Ziva basa rekunze \( f \) uye basa remukati \( g \):
– Basa rekunze: \( f(u) = \sin(u) \), uko \( u = g(x) \).
– Basa remukati: \( g(x) = x^2 \).
2. Tsvaga zvinobva pa \( f \) uye \( g \):
– \( f'(u) = \cos(u) \).
– \( g'(x) = 2x \).
3. Shandisa mutemo wecheni:
– \( h'(x) = f'(g(x)) \cdot g'(x) \).
– Saka, \( h'(x) = \cos(x^2) \cdot 2x \).
Saka, tine \( h'(x) = 2x \cos(x^2) \).
Kushandiswa kweMutemo weChain
Fizikisi
Mufizikisi, mutemo wecheni unowanzo shandiswa kuverenga velocity uye acceleration muma dynamical systems. Semuenzaniso, kana nzvimbo yechinhu ichi ikapihwa sebasa renguva \( s(t) \), uye nzvimbo iyoyo inoenderanawo neimwe variable yakaita setembiricha kana kumanikidzwa, tinogona kushandisa mutemo wecheni kuona hukama huripo pakati pevelocity kana acceleration yechinhu nechimwe chinhu ichocho.
upfumi
Munyaya dzezvehupfumi, kushandiswa kumwe kwemutemo wecheni kuri mukuongorora kwemari. Muchiitiko ichi, purofiti kana mitengo yekambani inogona kutsamira pane zvakasiyana-siyana, zvakaita semutengo wechigadzirwa, mitengo yekugadzira, kana huwandu hwakatengeswa. Mutemo wecheni unotibvumira kunzwisisa kuti shanduko mune chero chimwe chezviyero izvi zvinokanganisa sei purofiti kana mitengo yese.
Kusiyanisa Kwakajeka
Mutemo wecheni unokoshawo zvikuru mukusiyanisa zvinhu zvisina kujeka, uko tiri kubata nemabasa asina kutaurwa zvakajeka. Ngatitii tine equation \( x^2 + y^2 = 1 \) inomiririra denderedzwa reradius 1. Tinogona kushandisa mutemo wecheni kuwana derivative isina kujeka ye \( y \) maererano ne \( x \).
Ngatitorei mhedzisiro yemativi ese e equation:
\[ \frac{d}{dx} (x^2 + y^2) = \frac{d}{dx} (1) \]
Tichishandisa mutemo wecheni, tinowana:
\[ 2x + 2y \frac{dy}{dx} = 0 \]
Kugadzirisa \( \frac{dy}{dx} \):
\[ 2y \frac{dy}{dx} = -2x \]
\[ \frac{dy}{dx} = -\frac{x}{y} \]
Uyu muenzaniso wekuti mutemo wecheni unopa sei chishandiso chine simba chekusiyanisa zvisina kujeka muzviitiko apo \( y \) iri basa re \( x \) risingataurwi zvakajeka.
Mienzaniso Yakaoma
Ngatitii \( f(x) = e^{3x^2+2x} \). Tinoda kuwana chinobva pa \( f(x) \). Muchiitiko ichi, basa remukati ndi \( u(x) = 3x^2 + 2x \) uye basa rekunze ndi \( f(u) = e^u \).
Kushandisa mutemo wecheni:
1. Chinobva pabasa rekunze: \( f'(u) = e^u \).
2. Zvinobva pabasa remukati: \( u'(x) = 6x + 2 \).
Saka, zvichienderana nemutemo wecheni:
\[ f'(x) = e^{3x^2+2x} \cdot (6x+2) \]
Mhedziso
Mutemo wecheni chishandiso chakakosha mukuverenga kwakasiyana-siyana. Nekupa nzira yekuverenga derivative yecompound function, mutemo wecheni unowedzera huwandu hwekushandiswa kwezvinobva muzvikamu zvakasiyana-siyana, kubva kufizikisi kusvika kuhupfumi. Kuziva mutemo wecheni kwakakosha kwete chete pakunzwisisa pfungwa huru dzekuverenga asiwo pakushandisa matekiniki ekuverenga kumatambudziko chaiwo epasirese.
Mukudzidza nekushandisa mutemo wecheni, chinhu chikuru chekubudirira kunzwisisa chimiro chemabasa anobatanidzwa uye matanho chaiwo anodiwa kuti uishandise. Nekunzwisisa kwakasimba uye tsika dzakafanana, mutemo wecheni unogona kuva chishandiso chine simba mukugadzirisa matambudziko akasiyana-siyana emasvomhu uye mashandisirwo chaiwo.
