Muenzaniso wemubvunzo wekukurukurirana pamusoro pechinobva pabasa re algebraic
Chinobva mukuverenga (derivative) ipfungwa huru inoshandiswa kutsanangura kuti basa rinochinja sei, kana kuti kutsveyama kwebasa pane imwe nzvimbo. Zvinobva mumashoko zvinobatsira muzvikamu zvakasiyana-siyana zvakaita sefizikisi, economics, uye engineering nekuti zvinopa ruzivo nezve mwero wekuchinja. Muchinyorwa chino, tichakurukura mienzaniso yakati wandei yezvinobuda mumabasa e algebraic uye maitiro ekuzvigadzirisa.
Muenzaniso 1: Kubva paPolynomial Function
Mubvunzo: Zvichienderana nebasa \( f(x) = 3x^3 – 5x^2 + 2x – 7 \). Sarudza kuti chii chinobva mubasa racho!
Mhinduro:
Tichishandisa mutemo wekutanga wema derivatives emabasa e polynomial, kureva \(\frac{d}{dx} x^n = nx^{n-1} \), tichaverenga derivative yetemu yega yega yebasa rimwe nerimwe.
\[
\begin{align}
f(x) &= 3x^3 – 5x^2 + 2x – 7 \\
f'(x) &= \frac{d}{dx}(3x^3) – \frac{d}{dx}(5x^2) + \frac{d}{dx}(2x) – \frac{d}{dx}(7) \\
f'(x) &= 3 \cdot 3x^{3-1} – 5 \cdot 2x^{2-1} + 2 \cdot 1x^{1-1} – 0 \\
f'(x) &= 9x^2 – 10x + 2.
\end{align}
\]
Saka, chinobva pa \( f(x) = 3x^3 – 5x^2 + 2x – 7 \) ndi \( f'(x) = 9x^2 – 10x + 2 \).
Muenzaniso 2: Kubva paBasa rine Fractional Exponents
Mubvunzo: Tsvaga derivative yebasa \( g(x) = x^{3/2} + x^{1/2} \).
Mhinduro:
Uchishandisa mutemo mumwe chete wekutora, ndiwo \(\frac{d}{dx} x^n = nx^{n-1} \):
\[
\begin{align}
g(x) &= x^{3/2} + x^{1/2} \\
g'(x) &= \frac{d}{dx}(x^{3/2}) + \frac{d}{dx}(x^{1/2}) \\
g'(x) &= \frac{3}{2}x^{(3/2)-1} + \frac{1}{2}x^{(1/2)-1} \\
g'(x) &= \frac{3}{2}x^{1/2} + \frac{1}{2}x^{-1/2}.
\end{align}
\]
Saka, chinobva pa \( g(x) = x^{3/2} + x^{1/2} \) ndi \( g'(x) = \frac{3}{2}x^{1/2} + \frac{1}{2}x^{-1/2} \).
Muenzaniso 3: Zvibereko zveMabasa eExponential neTrigonometric
Mubvunzo: Sarudza chinobva pabasa \( h(x) = e^x \cdot \sin(x) \).
Mhinduro:
Kuti tigadzirise derivative iyi, tinoda Mutemo weChigadzirwa, unoti \((uv)' = u'v + uv'\). Ngatitii \( u(x) = e^x \) uye \( v(x) = \sin(x) \), zvino:
\[
\begin{align}
u'(x) &= e^x, & \text{nekuti chinobva pa } e^x \text{ ndi } e^x \\
v'(x) &= \cos(x), & \text{nekuti chinobva pa } \sin(x) \text{ ndi } \cos(x).
\end{align}
\]
Kushandisa mutemo wakabva kune zvigadzirwa:
\[
\begin{align}
h'(x) &= (e^x \cdot \sin(x))' \\
&= e^x \cdot (\sin(x))' + \sin(x) \cdot (e^x)' \\
&= e^x \cdot \cos(x) + \sin(x) \cdot e^x \\
&= e^x (\cos(x) + \sin(x)).
\end{align}
\]
Saka, chinobva pa \( h(x) = e^x \sin(x) \) ndi \( h'(x) = e^x (\cos(x) + \sin(x)) \).
Muenzaniso 4: Kubva paBasa Uchishandisa Mutemo weChain
Mubvunzo: Tsvaga chinobva pabasa \( k(x) = (3x^2 – x + 4)^5 \).
Mhinduro:
Kuti tigadzirise derivative iyi, tinoda mutemo wecheni, unoti \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\). Ngatitii \( u(x) = 3x^2 – x + 4 \) uye \( f(u) = u^5 \), zvino:
\[
\begin{align}
k(x) &= (3x^2 – x + 4)^5 \\
u(x) &= 3x^2 – x + 4, uye \text{so} \\
k(x) &= f(u(x)) = u^5 \\
k'(x) &= 5u^4 \cdot u'(x) \\
u'(x) &= \frac{d}{dx}(3x^2 – x + 4) \\
&= 6x – 1.
\end{align}
\]
Kushandisa mutemo wecheni:
\[
\begin{align}
k'(x) &= 5(3x^2 – x + 4)^4 \cdot (6x – 1) \\
&= 5(3x^2 – x + 4)^4 (6x – 1).
\end{align}
\]
Saka, chinobva pa \( k(x) = (3x^2 – x + 4)^5 \) ndi \( k'(x) = 5 (3x^2 – x + 4)^4 (6x – 1) \).
Muenzaniso 5: Kubva paBasa rine Trigonometric Identities
Mubvunzo: Sarudza chinobva pabasa \( m(x) = \sin(x) \cdot \cos(x) \).
Mhinduro:
Tichashandisa mutemo wekubvisa zvinhu (derivative rule) pazvigadzirwa. Ngatitii \( u(x) = \sin(x) \) uye \( v(x) = \cos(x) \), zvino:
\[
\begin{align}
u'(x) &= \cos(x), \\
v'(x) &= -\chivi(x).
\end{align}
\]
Kushandisa mutemo wakabva kune zvigadzirwa:
\[
\begin{align}
m'(x) &= (\sin(x) \cdot \cos(x))' \\
&= (\chivi(x))' \cdot \cos(x) + \chivi(x) \cdot (\cos(x))' \\
&= \cos(x) \cdot \cos(x) + \sin(x) \cdot (-\sin(x)) \\
&= \cos^2(x) – \chivi^2(x).
\end{align}
\]
Kushandisa trigonometric identity \(\cos(2x) = \cos^2(x) – \sin^2(x)\):
\[
m'(x) = \cos(2x).
\]
Saka, chinobva pa \( m(x) = \sin(x) \cdot \cos(x) \) ndi \( m'(x) = \cos(2x) \).
Mhedziso
Chinobva pabasa re algebraic ipfungwa huru mu calculus inokosha zvikuru uye inobatsira mukushandiswa kwakasiyana-siyana. Mitemo yakasiyana-siyana yekubuda, yakadai semutemo wekutanga we derivative, mutemo wechigadzirwa, mutemo we chain, uye mitemo ye trigonometric derivatives, zvese zvinobatsira pakuverenga ma derivatives emabasa akaomarara. Nekunzwisisa mienzaniso iri pamusoro uye kuita matambudziko, tinogona kuvandudza kunzwisisa kwedu nehunyanzvi mukutora ma derivatives emabasa e algebraic.
