Mibvunzo yeMienzaniso neKukurukurirana kweChirevo cheMabasa Anobva Pabasa
Chinobva pabasa (derivative of a function) ipfungwa huru mukuverenga ine mashandisirwo akawanda muzvidzidzo zvakasiyana-siyana, zvakaita sefizikisi, economics, uye engineering. Chinyorwa chino chichataura nezvemienzaniso yakati wandei yematambudziko uye chichakurukura pfungwa yekubva pabasa (derivative of a function) kuti tinzwisise zvakadzama nyaya iyi.
Tsanangudzo Yekutanga yeZvinobva Mumashoko
Tisati tapinda mumibvunzo yemuenzaniso, zvakanaka kuti tiongorore muchidimbu tsananguro uye hwaro hwema derivatives. Derivative yebasa \( f(x) \) panzvimbo \( x = a \) ndeiyi:
\[ f'(a) = \lim_{{h \to 0}} \frac{f(a+h) – f(a)}{h} \]
Basa \( f'(x) \) rinonzi basa rekubva pa \( f(x) \).
Muenzaniso Mubvunzo 1: Zvibereko zvePolynomial zveBasic
Mubvunzo:
Tsvaga derivative yekutanga yebasa \( f(x) = 3x^3 – 5x^2 + 2x – 7 \).
Kukurukurirana:
Shandisa mutemo wekutanga wekubvisa \( \frac{d}{dx} x^n = nx^{n-1} \).
1. Kune \( 3x^3 \):
\[ \frac{d}{dx}(3x^3) = 3 \cdot 3x^{3-1} = 9x^2 \]
2. Kune \( -5x^2 \):
\[ \frac{d}{dx}(-5x^2) = -5 \cdot 2x^{2-1} = -10x \]
3. Kune \( 2x \):
\[ \frac{d}{dx}(2x) = 2 \]
4. Kune \( -7 \):
\[ \frac{d}{dx}(-7) = 0 \]
Saka:
\[ f'(x) = 9x^2 – 10x + 2 \]
Muenzaniso Mubvunzo 2: Zvibereko zveMabasa eTrigonometric
Mubvunzo:
Tsvaga derivative yekutanga yebasa \( g(x) = \sin(x) \cdot \cos(x) \).
Kukurukurirana:
Shandisa mutemo wechigadzirwa \( \frac{d}{dx} [u(x) \cdot v(x)] = u'(x)v(x) + u(x)v'(x) \) na \( u(x) = \sin(x) \) uye \( v(x) = \cos(x) \).
1. Chinobva pa \( \sin(x) \) ndi \( \cos(x) \), saka \( u'(x) = \cos(x) \).
2. Chinobva pa \( \cos(x) \) ndi \( -\sin(x) \), saka \( v'(x) = -\sin(x) \).
Kutsiva \( u'(x) \) uye \( v'(x) \):
\[ g'(x) = \cos(x) \cdot \cos(x) + \sin(x) \cdot (-\sin(x)) \]
\[ g'(x) = \cos^2(x) – \sin^2(x) \]
Mhedzisiro yekupedzisira:
\[ g'(x) = \cos^2(x) – \sin^2(x) \]
Muenzaniso 3: Kubva paBasa reExponential
Mubvunzo:
Tsvaga derivative yekutanga yebasa \( h(x) = e^{2x} \).
Kukurukurirana:
Shandisa mutemo we derivative ye exponential function \( \frac{d}{dx} e^{kx} = ke^{kx} \) na \( k = 2 \).
\[ h'(x) = \frac{d}{dx} e^{2x} \]
\[ h'(x) = 2 \cdot e^{2x} \]
Mhedzisiro yekupedzisira:
\[ h'(x) = 2e^{2x} \]
Muenzaniso Mubvunzo 4: Kubva paLogarithmic Function
Mubvunzo:
Tsvaga derivative yekutanga yebasa \( p(x) = \ln(3x + 1) \).
Kukurukurirana:
Shandisa mutemo we derivative yebasa re logarithmic \( \frac{d}{dx} \ln(u) = \frac{1}{u} \cdot u' \) na \( u(x) = 3x + 1 \).
1. Tsvaga chinhu chinobva mukati \( u(x) = 3x + 1 \):
\[ u'(x) = 3 \]
2. Shandisa mutemo we logarithmic derivative:
\[ p'(x) = \frac{1}{3x + 1} \cdot 3 \]
Mhedzisiro yekupedzisira:
\[ p'(x) = \frac{3}{3x + 1} \]
Muenzaniso Mubvunzo 5: Kushandiswa kweZvinobva muMashoko - Zvakanyanya uye Zvishoma
Mubvunzo:
Tsvaga manhamba epamusoro uye epasi ebasa \( q(x) = -2x^3 + 3x^2 + 12x – 5 \) pane imwe nguva \( x \in [-2, 2] \).
Kukurukurirana:
1. Tsvaga derivative yekutanga ye \( q(x) \):
\[ q'(x) = \frac{d}{dx}(-2x^3 + 3x^2 + 12x – 5) \]
\[ q'(x) = -6x^2 + 6x + 12 \]
2. Tsvaga nzvimbo dzisingachinji nekugadzirisa \( q'(x) = 0 \):
\[ -6x^2 + 6x + 12 = 0 \]
\[ -6(x^2 – x – 2) = 0 \]
\[ x^2 – x – 2 = 0 \]
\[ (x-2)(x+1) = 0 \]
Nzvimbo dzisingachinji ndi \( x = 2 \) uye \( x = -1 \).
3. Ongorora \( q(x) \) panzvimbo dzakakosha uye miganhu yepakati:
\[ q(-2) = -2(-2)^3 + 3(-2)^2 + 12(-2) – 5 \]
\[ = 16 + 12 – 24 – 5 \]
\[ = -1 \]
\[ q(2) = -2(2)^3 + 3(2)^2 + 12(2) – 5 \]
\[ = -16 + 12 + 24 – 5 \]
\[ = 15 \]
\[ q(-1) = -2(-1)^3 + 3(-1)^2 + 12(-1) – 5 \]
\[ = 2 + 3 – 12 – 5 \]
\[ = -12 \]
4. Kuongororwa kwemigumisiro:
– Kukosha kukuru kunoitika pa \( x = 2 \) ne \( q(2) = 15 \).
– Kukosha kushoma kunoitika pa \( x = -1 \) ne \( q(-1) = -12 \).
Penutup
Kunzwisisa zvakakwana pfungwa yekubva kune chimwe chinhu (derivative) chebasa kwakakosha munzvimbo dzakasiyana dzesainzi. Tinovimba kuti mienzaniso yematambudziko nehurukuro dziri pamusoro apa zvichabatsira kuwedzera kunzwisisa kwako pfungwa yacho. Mukuita, tinowanzo fanira kusanganisa mitemo yakasiyana-siyana nedzidziso kuti tigadzirise matambudziko akaomarara. Kudzidza kwakanaka!
