Kunyora Derivative yeBasa
Pendauluan
Mumasvomhu, kunyanya calculus, derivative ipfungwa huru ine basa rakakosha mukushandiswa kwakasiyana-siyana. Maderivatives haangoshandiswi muzvidzidzo zvemasvomhu chete asiwo mune zvesainzi, engineering, economics, nedzimwe nzvimbo dzakawanda. Chinyorwa chino chichakurukura derivative yebasa zvakadzama, ichifukidza hwaro hwaro, mitemo yakakosha, uye mienzaniso yekushandiswa.
Nheyo dzeZvibereko
Tsanangudzo yeZvinobva muMashoko
Chinobva pabasa chinotsanangura mwero wekuchinja kwebasa maererano nechinhu charo chakazvimiririra. Nenzira yekufungidzira, chinobva pabasa chinogona kutsanangurwa senzira yekutsveyama kwemutsetse wetangent unobata girafu yebasa pane imwe nzvimbo.
Kana \( y = f(x) \), ipapo derivative yekutanga ye \( f \) maererano ne \( x \) inoratidzirwa ne \( f'(x) \) kana \( \frac{dy}{dx} \). Tsanangudzo yepamutemo yederivative inopiwa nemuganhu unotevera:
\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) – f(x)}{h} \]
Runyoro Rwakabva
Kune mamwe mashoko anowanzo shandiswa pakunyora mashoko anobva pashoko:
1. Mashoko aLeibniz: \( \frac{dy}{dx} \)
2. Lagrange notation: \( f'(x) \)
3. Mashoko aNewton: \( y' \)
4. Kunyorwa kwaEuler: \( Df(x) \)
Chinyorwa chimwe nechimwe chine mashandisirwo chaiwo uye mamiriro ezvinhu anoshandiswa zvakanyanya.
Mitemo Yekutanga Mukusiyanisa
Mitemo yekuwedzera nekubvisa
Kana \( f(x) \) uye \( g(x) \) ari mabasa maviri anogona kupatsanurwa, saka:
\[ \frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x) \]
Mitemo yekuwedzera
Kune mabasa maviri \( u(x) \) uye \( v(x) \):
\[ \frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) \]
Mitemo yeDivision
Kana \( u(x) \) uye \( v(x) \) ari mabasa maviri, uye \( v(x) \neq 0 \):
\[ \frac{d}{dx} \left[ \frac{u(x)}{v(x)} \right] = \frac{u'(x) \cdot v(x) – u(x) \cdot v'(x)}{[v(x)]^2} \]
Mutemo weCheni
Pakuumbwa kwemabasa maviri \( f(u) \) uye \( u(g) \):
\[ \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) \]
Mienzaniso yeMashandisirwo
Zvibereko zveMabasa ePolynomial
Ngatitii \( f(x) = 3x^3 – 5x^2 + 2x – 1 \). Kuti tiwane derivative yebasa iri, tinoshandisa mitemo yekutanga yekusiyanisa.
\[ f'(x) = \frac{d}{dx} (3x^3) – \frac{d}{dx} (5x^2) + \frac{d}{dx} (2x) – \frac{d}{dx} (1) \]
\[ f'(x) = 9x^2 – 10x + 2 \]
Zvibereko zveMabasa eExponential neLogarithmic
Kana \( f(x) = e^x \), saka derivative yebasa re exponential ndeiyi:
\[ f'(x) = e^x \]
Pabasa re "natural logarithm" \( f(x) = \ln(x) \):
\[ f'(x) = \frac{1}{x} \]
Zvibereko zveMabasa eTrigonometric
Kune mabasa ekutanga etrigonometric:
– Kana \( f(x) = \sin(x) \), saka \( f'(x) = \cos(x) \)
– Kana \( f(x) = \cos(x) \), saka \( f'(x) = -\sin(x) \)
– Kana \( f(x) = \tan(x) \), saka \( f'(x) = \sec^2(x) \)
Kubva paComposite Function
Ngatitii \( f(x) = \sin(2x) \). Tinogona kushandisa mutemo wecheni:
\[ f'(x) = \cos(2x) \cdot \frac{d}{dx}(2x) = \cos(2x) \cdot 2 = 2\cos(2x) \]
Zvigadzirwa Zvepamusoro
Zvibereko zveChipiri neZvinotevera
Chinobva pachikamu chechipiri ndicho chinobva pachikamu chekutanga chebasa rekubva pachikamu. Kana \( y = f(x) \) ipapo chinobva pachikamu chechipiri chinoratidzwa ne \( f”(x) \) kana \( \frac{d^2y}{dx^2} \). Uye zvichingodaro kune chinobva pachikamu chechitatu \( f”'(x) \) kana \( \frac{d^3y}{dx^3} \).
Ngatitii \( f(x) = x^4 \):
\[ f'(x) = 4x^3 \]
\[ f”(x) = \frac{d}{dx}(4x^3) = 12x^2 \]
\[ f”'(x) = \frac{d}{dx}(12x^2) = 24x \]
\[ f””(x) = \frac{d}{dx}(24x) = 24 \]
Mashandisirwo eZvinobva muFizikisi
Mufizikisi, maderivatives anowanzo shandiswa kuona velocity uye acceleration. Ngatitii \( s(t) \) ibasa renzvimbo maererano nenguva \(t \). Velocity \(v(t) \) ndiyo derivative yekutanga yenzvimbo:
\[ v(t) = s'(t) \]
Kukurumidza \( a(t) \) ndiyo derivative yekutanga yevelocity kana derivative yechipiri yenzvimbo:
\[ a(t) = v'(t) = s”(t) \]
Mhedziso
Chinobva pabasa ipfungwa huru mukuverenga nekushandisa kwakapararira munzvimbo dzakasiyana-siyana. Kunzwisisa zviri nyore kwekubva pabasa senzira yekutsvedza kwemutsetse une tangent kunopa ruzivo rwakakosha pamusoro pehunhu uye maitiro ebasa. Kunzwisisa nekukwanisa kushandisa mitemo yekusiyanisa senge mutemo wecheni, mutemo wechigadzirwa, uye mutemo wekupatsanura zvakakosha kune chero munhu ari kudzidza kuverenga nekushandisa. Kuburikidza nemienzaniso iri nyore uye mashandisirwo mufizikisi, chinyorwa chino chinotarisira kupa kunzwisisa kwakazara kwekunyora kubva pabasa.
