Mienzaniso yeMibvunzo Inokurukura NezveKunyora Mashandiro Ebasa
Chinobva pabasa (derivative of a function) ipfungwa huru mukuverenga, inowanzoshandiswa muzvikamu zvakasiyana-siyana zvesainzi, zvakaita sefizikisi, economics, biology, uye engineering. Chinobva pabasa (derivative of a function) chinoyera kuti kukosha kwaro kunochinja nekukurumidza sei maererano nekuchinja kwezvimiro zvaro zvakazvimiririra. Muchinyorwa chino, tichakurukura mienzaniso yakati wandei yezvinetso zvinosanganisira kunyora chinobva pabasa (derivative of a function), pamwe netsananguro.
Muenzaniso Mubvunzo 1: Kubva paMabasa Akareruka
Mubvunzo: Tsvaga derivative yekutanga yebasa \( f(x) = 3x^2 + 5x + 7 \).
Kukurukurirana:
Kuti tizive derivative yekutanga yebasa \( f(x) \), tinoshandisa mitemo yekutanga yekusiyanisa, inoti:
\[
\frac{d}{dx}(ax^n) = anx^{n-1}
\]
Saka, tinogona kuverenga derivative yezwi rega rega mubasa iri seinotevera:
\[
f'(x) = \frac{d}{dx}(3x^2) + \frac{d}{dx}(5x) + \frac{d}{dx}(7)
\]
\[
f'(x) = 3 \cdot 2x^{2-1} + 5 \cdot 1x^{1-1} + 0
\]
\[
f'(x) = 6x + 5
\]
Saka, derivative yekutanga yebasa \( f(x) = 3x^2 + 5x + 7 \) ndi \( f'(x) = 6x + 5 \).
Muenzaniso Mubvunzo 2: Zvibereko zveMabasa eTrigonometric
Mubvunzo: Tsvaga derivative yekutanga yebasa \( g(x) = \sin(x) + \cos(x) \).
Kukurukurirana:
Tinoshandisa mitemo yekutanga yekubvisa ma "trigonometric functions":
\[
\frac{d}{dx}(\sin(x)) = \cos(x)
\]
\[
\frac{d}{dx}(\cos(x)) = -\sin(x)
\]
Saka:
\[
g'(x) = \frac{d}{dx}(\sin(x)) + \frac{d}{dx}(\cos(x))
\]
\[
g'(x) = \cos(x) – \sin(x)
\]
Saka, derivative yekutanga yebasa \( g(x) = \sin(x) + \cos(x) \) ndi \( g'(x) = \cos(x) – \sin(x) \).
Muenzaniso Mubvunzo 3: Basa reKuwedzera
Mubvunzo: Tsvaga derivative yekutanga yebasa \( h(x) = x^2 \sin(x) \).
Kukurukurirana:
Kune mabasa ari chibereko chemabasa maviri, tinoshandisa mutemo wekuwedzera:
\[
\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
\]
Ngatitii \( u(x) = x^2 \) uye \( v(x) = \sin(x) \). Zvadaro:
\[
u'(x) = \frac{d}{dx}(x^2) = 2x
\]
\[
v'(x) = \frac{d}{dx}(\sin(x)) = \cos(x)
\]
Tichishandisa mutemo wekuwedzera, tinogona kunyora:
\[
h'(x) = [x^2]' \chivi(x) + x^2 [\chivi(x)]'
\]
\[
h'(x) = 2x \chivi(x) + x^2 \cos(x)
\]
Saka, derivative yekutanga yebasa \( h(x) = x^2 \sin(x) \) ndi \( h'(x) = 2x \sin(x) + x^2 \cos(x) \).
Muenzaniso Mubvunzo 4: Kubva paBasa reKuumba
Mubvunzo: Tsvaga derivative yekutanga yebasa \( k(x) = \sin(x^2) \).
Kukurukurirana:
Kune mabasa ari musanganiswa wemabasa maviri, tinoshandisa mutemo wecheni:
\[
\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)
\]
Regai \( f(u) = \sin(u) \) uye \( u = x^2 \). Zvadaro \( f'(u) = \cos(u) \) uye \( g'(x) = \frac{d}{dx}(x^2) = 2x \).
Tichishandisa mutemo wecheni, tinogona kunyora kuti:
\[
k'(x) = \frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x
\]
Saka, derivative yekutanga yebasa \( k(x) = \sin(x^2) \) ndi \( k'(x) = 2x \cos(x^2) \).
Muenzaniso Mubvunzo 5: Kubva kuMabasa Akanaka
Dambudziko: Tsvaga derivative yekutanga yebasa \( m(x) = \frac{2x}{x^2 + 1} \).
Kukurukurirana:
Kune mabasa ari quotient yema function maviri, tinoshandisa mutemo we quotient:
\[
\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) – u(x)v'(x)}{[v(x)]^2}
\]
Ngatitii \( u(x) = 2x \) uye \( v(x) = x^2 + 1 \). Zvadaro:
\[
u'(x) = 2
\]
\[
v'(x) = \frac{d}{dx}(x^2 + 1) = 2x
\]
Tichishandisa mutemo we quotient, tinogona kunyora:
\[
m'(x) = \frac{[2x]'(x^2 + 1) – 2x[x^2 + 1]'}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2(x^2 + 1) – 2x \cdot 2x}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2x^2 + 2 – 4x^2}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{-(2x^2 – 2)}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2 – 2x^2}{(x^2 + 1)^2}
\]
Saka, derivative yekutanga yebasa \( m(x) = \frac{2x}{x^2 + 1} \) ndi \( m'(x) = \frac{2 – 2x^2}{(x^2 + 1)^2} \).
Mhedziso
Muchinyorwa chino, takurukura mienzaniso yakati wandei yematambudziko anosanganisira ma derivatives emabasa, kubva kumabasa akareruka, mabasa e trigonometric, kuwanda, kuumbwa, uye mabasa ane pfungwa. Muenzaniso wega wega unoratidza kushandiswa kwakakodzera kwemitemo ye derivative, senge mutemo wekutanga, mutemo wecheni, mutemo we kuwanda, uye mutemo we quotient. Kunzwisisa mashandisirwo emitemo iyi kwakakosha pakugadzirisa matambudziko akaomarara ekuverenga muzvikamu zvakasiyana-siyana. Kudzidzira nekudzidziswa kakawanda kuchabatsira kusimbisa kunzwisisa kwako nehunyanzvi hwako mukusiyanisa mabasa.
