Mibvunzo yemuenzaniso nekukurukurirana kwehunhu hwemabasa anobva kune mamwe marudzi
Chinobva pabasa (derivative of a function) ipfungwa huru mukuverenga inobatsira zvikuru pakuongorora maitiro emamwe mabasa. Muchinyorwa chino, tichakurukura mienzaniso yakati wandei yematambudziko uye tichakurukura nezvehunhu hwechinobva pabasa (derivative of a function).
Nhanganyaya kuBasa Derivatives
Chinobva pabasa \( f \) chinoratidzwa se \( f'(x) \). Chinobva pabasa chekutanga chinopa mwero wekuchinja kwebasa maererano nekuchinja kwaro kwakazvimirira. Rimwe izwi rinowanzoshandiswa ndiro differential. Kana \( y = f(x) \), saka chinobva pabasa \( f \) maererano ne \( x \) ndeichi:
\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) – f(x)}{h} \]
Hunhu hweZvinobva paBasa
Zvimwe zvinhu zvakakosha zve derivative yebasa ndeizvi:
1. Kurongeka: Kana \( f(x) \) uye \( g(x) \) ari mabasa anogovaniswa, uye \( c \) ari chinhu chisingachinji, saka:
\[
\frac{d}{dx} [cf(x) + g(x)] = c f'(x) + g'(x)
\]
2. Mutemo weChain: Yebasa rekubatanidza \( g(f(x)) \):
\[
\frac{d}{dx} g(f(x)) = g'(f(x)) \cdot f'(x)
\]
3. Chigadzirwa: Pamabasa \( u(x) \) uye \( v(x) \):
\[
\frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]
4. Quotient : Pamabasa \( u(x) \) uye \( v(x) \) apo \( v(x) \neq 0 \):
\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) – u(x)v'(x)}{(v(x))^2}
\]
Mibvunzo yemuenzaniso nekukurukurirana
Muenzaniso 1: Kuziva Zvinobva paBasa Rakareruka
Ngatitii \( f(x) = 3x^2 + 5x – 4 \). Sarudza derivative yebasa.
Mhinduro:
Tichashandisa mitemo yekutanga yekusiyanisa.
\[
f(x) = 3x^2 + 5x – 4
\]
Chinobva chekutanga:
\[
f'(x) = \frac{d}{dx} (3x^2) + \frac{d}{dx} (5x) – \frac{d}{dx} (4)
\]
Kuverenga chimwe nechimwe chinobva:
\[
\frac{d}{dx} (3x^2) = 6x
\]
\[
\frac{d}{dx} (5x) = 5
\]
\[
\frac{d}{dx} (4) = 0
\]
Kuti:
\[
f'(x) = 6x + 5
\]
Muenzaniso 2: Kushandisa Mutemo weChain
Zvichienderana nebasa \( y = (2x^3 – x^2 + 1)^5 \). Sarudza kuti chii chinobva pabasa racho.
Mhinduro:
Shandisa mutemo wecheni. Ngatitii \( u = 2x^3 – x^2 + 1 \), ipapo basa rinogona kunyorwazve se \( y = u^5 \).
Kutanga, tsvaga chinobva pa \( y \) maererano ne \( u \):
\[
\frac{dy}{du} = 5u^4
\]
Tevere, tsvaga chinobva pa \( u \) maererano ne \( x \):
\[
u = 2x^3 – x^2 + 1
\]
\[
\frac{du}{dx} = 6x^2 – 2x
\]
Sanganisa ma derivatives maviri aya nemutemo we chain:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 5u^4 \cdot (6x^2 – 2x)
\]
Tsiva zvakare \( u = 2x^3 – x^2 + 1 \):
\[
\frac{dy}{dx} = 5(2x^3 – x^2 + 1)^4 \cdot (6x^2 – 2x)
\]
Muenzaniso 3: Kushandisa Mitemo Yechigadzirwa
Zvichipiwa \( f(x) = x^2 e^x \). Sarudza derivative yebasa.
Mhinduro:
Shandisa mutemo wechigadzirwa, kureva kuti, kana \( u(x) = x^2 \) uye \( v(x) = e^x \), zvino:
\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\]
Kutanga, verenga zvinobuda pa \( u(x) \) uye \( v(x) \):
\[
u(x) = x^2 \zvinoreva u'(x) = 2x
\]
\[
v(x) = e^x \zvinoreva v'(x) = e^x
\]
Nekushandisa mitemo yechigadzirwa:
\[
f'(x) = 2x \cdot e^x + x^2 \cdot e^x = e^x (2x + x^2)
\]
Muenzaniso 4: Kushandisa Mutemo weQuotient
Zvapiwa \( f(x) = \frac{x^2 + 1}{x + 2} \). Tsvaga derivative yebasa.
Mhinduro:
Shandisa mutemo we quotient, kureva kuti kana \( u(x) = x^2 + 1 \) uye \( v(x) = x + 2 \), zvino:
\[
f'(x) = \frac{u'(x)v(x) – u(x)v'(x)}{[v(x)]^2}
\]
Kutanga, verenga zvinobuda pa \( u(x) \) uye \( v(x) \):
\[
u(x) = x^2 + 1 \zvinoreva u'(x) = 2x
\]
\[
v(x) = x + 2 \zvinoreva v'(x) = 1
\]
Nekushandisa mutemo we quotient:
\[
f'(x) = \frac{2x(x + 2) – (x^2 + 1)(1)}{(x + 2)^2}
\]
\[
f'(x) = \frac{2x^2 + 4x – x^2 – 1}{(x + 2)^2}
\]
\[
f'(x) = \frac{x^2 + 4x – 1}{(x + 2)^2}
\]
Mhedziso
Mukuverenga, kunzwisisa pfungwa huru yezvinobuda muzvikamu (derivatives) uye hunhu hwazvo kwakakosha pakugadzirisa matambudziko akasiyana-siyana emasvomhu. Chinyorwa chino chinotsanangura nzira dzakasiyana-siyana dzekuwana mabasa nekuratidza kushandiswa kwemitemo mikuru yakaita semutsara, cheni, zvigadzirwa, uye maquotients kuburikidza nemienzaniso yakati wandei nekukurukurirana kwakadzama. Nekunzwisisa uye kugara tichiita zvinobuda muzvikamu (derivatives), tinogona kuva nehunyanzvi mukuongorora shanduko mumabasa munzvimbo dzakasiyana-siyana.
