Mibvunzo yemuenzaniso nekukurukurirana kweMutemo weChain muDerivatives
Mutemo wecheni ndeimwe yepfungwa dzakakosha mu differential calculus, inoshandiswa kuverenga derivative yebasa rine mabasa maviri kana anopfuura. Muchinyorwa chino, tichakurukura pfungwa huru yemutemo wecheni, mashandisirwo awo, uye mienzaniso yekushandiswa kwawo mumatambudziko e derivative anowanzo itika kuchikoro chesekondari nekukoreji.
1. Nhanganyaya kuMutemo weCheni
Tisati tapinda mudambudziko remuenzaniso, ngatitangei tanzwisisa kuti mutemo wecheni chii. Mutemo wecheni unoti kana tine mabasa maviri akasiyana \( f \) uye \( g \), uye tinoda kuwana derivative yemusanganiswa wemabasa \( h = f(g(x)) \), saka derivative ye \( h \) ndeiyi:
\[ h'(x) = f'(g(x)) \cdot g'(x) \]
Muchidimbu, tinoverenga derivative yebasa rekunze pa g(x), tobva tawedzera mhedzisiro ne derivative yebasa remukati \( g(x) \).
2. Kunzwisisa Basa reKuumbwa
Tisati tapinda mumatambudziko emuenzaniso, zvakakosha kunzwisisa mabasa ekunyora. Basa rekunyora ibasa rinowanikwa nekuisa basa rimwe mune rimwe. Semuenzaniso, kana tiine \( f(x) = \sin(x) \) uye \( g(x) = x^2 \), saka kuumbwa kwebasa mbiri idzi kuchava \( h(x) = f(g(x)) = \sin(x^2) \).
Mumabasa ekuumbwa, tinowanzo funga nezve \( g(x) \) se "basa remukati" uye \( f(x) \) se "basa rekunze". Mumuenzaniso uyu, basa remukati ndi \( x^2 \) uye basa rekunze ndi sine.
3. Mibvunzo yemuenzaniso nekukurukurirana
Ngatitarisei mimwe mienzaniso yematambudziko anoshandisa mutemo wecheni kuzvigadzirisa.
Muenzaniso wechishanu:
Zvichienderana nebasa \( y = \cos(3x^2) \), tsvaga derivative yekutanga ya y maererano na x.
Kukurukurirana:
Kutanga, tinoona mabasa emukati neekunze. Pano, basa remukati ndi \( g(x) = 3x^2 \) uye basa rekunze ndi \( f(g) = \cos(g) \).
Tinoziva:
1. \( g'(x) = 6x \)
2. \( f'(g) = -\chivi(g) \)
Nemutemo wecheni, tinowana:
\[ y' = f'(g(x)) \cdot g'(x) = -\sin(3x^2) \cdot 6x \]
Saka, chinobva pa \( y = \cos(3x^2) \) ndeichi:
\[ y' = -6x \chivi(3x^2) \]
Muenzaniso wechishanu:
Tsvaga derivative yekutanga ye \( h(x) = e^{5x^3 + 2x} \).
Kukurukurirana:
Pano basa remukati ndi \( g(x) = 5x^3 + 2x \) uye basa rekunze ndi \( f(g) = e^g \).
Tinoziva:
1. \( g'(x) = 15x^2 + 2 \)
2. \( f'(g) = e^g \)
Nemutemo wecheni, tinowana:
\[ h'(x) = f'(g(x)) \cdot g'(x) = e^{5x^3 + 2x} \cdot (15x^2 + 2) \]
Saka, chinobva pa \( h(x) = e^{5x^3 + 2x} \) ndeichi:
\[ h'(x) = (15x^2 + 2)e^{5x^3 + 2x} \]
Muenzaniso wechishanu:
Tsvaga derivative yekutanga ye \( y = \ln(4x^2 - 5) \).
Kukurukurirana:
Basa remukati ndi \( g(x) = 4x^2 – 5 \) uye basa rekunze ndi \( f(g) = \ln(g) \).
Tinoziva:
1. \( g'(x) = 8x \)
2. \( f'(g) = \frac{1}{g} \)
Nemutemo wecheni, tinowana:
\[ y' = f'(g(x)) \cdot g'(x) = \frac{1}{4x^2 – 5} \cdot 8x \]
Saka, chinobva pa \( y = \ln(4x^2 – 5) \) ndeichi:
\[ y' = \frac{8x}{4x^2 – 5} \]
Muenzaniso wechishanu:
Zvichienderana nebasa \( y = (3x^2 + 2x + 1)^4 \), tsvaga derivative yayo.
Kukurukurirana:
Basa remukati ndi \( g(x) = 3x^2 + 2x + 1 \) uye basa rekunze ndi \( f(g) = g^4 \).
Tinoziva:
1. \( g'(x) = 6x + 2 \)
2. \( f'(g) = 4g^3 \)
Nemutemo wecheni, tinowana:
\[ y' = f'(g(x)) \cdot g'(x) = 4(3x^2 + 2x + 1)^3 \cdot (6x + 2) \]
Saka, chinobva pa \( y = (3x^2 + 2x + 1)^4 \) ndeichi:
\[ y' = 4(3x^2 + 2x + 1)^3 (6x + 2) \]
4. Nyaya Dzakakosha uye Kugadzirwa kweMitemo yeCheni
Dzimwe nguva, mutemo wecheni haugumiri pakuumbwa kwemafunction maviri chete. Pane dzimwe nguva apo basa riri kuumbwa kwemafunction anopfuura maviri, semuenzaniso: \( h(x) = f(g(k(x))) \).
Panyaya yemabasa matatu, mutemo wecheni unogona kushandiswa muzvikamu:
\[ h'(x) = f'(g(k(x))) \cdot g'(k(x)) \cdot k'(x) \]
Tinogona kuona kuti muchikamu chimwe nechimwe, tinoverenga zvinobuda muzvikamu zvekunze tisati taenderera mberi kune zvinobuda muzvikamu zvemukati.
Muenzaniso wechishanu:
Zvichipiwa \( y = \sqrt{\ln(2x^2 + 1)} \), tsvaga chinobva pairi.
Kukurukurirana:
Basa remukati-kati ndi \( k = 2x^2 + 1 \), pakati: \( g = \ln(k) \) uye rekunze: \( f = \sqrt{g} \).
Tinoziva:
1. \( k'(x) = 4x \)
2. \( g'(k) = \frac{1}{k} \)
3. \( f'(g) = \frac{1}{2\sqrt{g}} \)
Ngatishandise mutemo wecheni muzvikamu:
\[ y' = f'(g(k(x))) \cdot g'(k(x)) \cdot k'(x) = \frac{1}{2\sqrt{\ln(2x^2 + 1)}} \cdot \frac{1}{2x^2 + 1} \cdot 4x \]
Saka chinobva pa \( y = \sqrt{\ln(2x^2 + 1)} \) ndeichi:
\[ y' = \frac{4x}{2(2x^2 + 1)\sqrt{\ln(2x^2 + 1)}} \]
5. Kesimpulan
Mutemo wecheni une basa rakakosha mukuverenga kwakasiyana-siyana, kunyanya kana uchibata nekuumbwa kwemabasa. Kunzwisisa uye kugona mutemo wecheni kunopa hwaro hwakasimba hwekugadzirisa matambudziko akaomarara mukuverenga. Chinyorwa chino chakurukura mienzaniso yakakosha yakati wandei kuti tinzwisise kushandiswa kwemutemo wecheni kune zvinobva muzvinyorwa. Tinovimba kuti hurukuro iyi inobatsira vadzidzi uye inogona kushandiswa mumamiriro akasiyana-siyana emasvomhu akaoma.
