Tusiaina o le Fa'asologa o se Galuega Fa'atino
Pendahuluan
I le matematika, aemaise lava le calculus, o le derivative o se manatu faavae e taua tele i le tele o faʻaoga. E le gata ina faʻaaogaina derivatives i le matematika faʻateorē ae faʻapea foʻi i le saienisi, inisinia, tamaoaiga, ma le tele o isi matāʻupu. O lenei tusiga o le a talanoaina auiliili ai le derivative o se galuega faatino, e aofia ai ona faʻavae, tulafono taua, ma faʻataʻitaʻiga o faʻaoga.
Fa'avae o Mea e Maua Mai Fa'asili
Fa'amatalaga o Mea e Maua Mai
O le derivative o se galuega faatino e faʻamatalaina ai le fua faatatau o le suiga o le galuega faatino e faʻatatau i lona fesuiaʻiga tutoʻatasi. I se tulaga faʻapitoa, e mafai ona faʻamatalaina le derivative o le faʻasolo o le laina tangent e paʻi atu i le kalafi o le galuega faatino i se tulaga.
Afai o le \( y = f(x) \), o lona uiga o le ulua'i fa'aliliuga o le \( f \) e fa'atatau i le \( x \) ua fa'ailoa mai e le \( f'(x) \) po'o le \( \frac{dy}{dx} \). O le fa'amatalaga aloa'ia o le fa'aliliuga e tu'uina atu e le tapula'a lenei:
\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) – f(x)}{h} \]
Fa'ailoga Fa'atupu
E tele naua fa'ailoga e masani ona fa'aaogaina i le tusiaina o fa'ailoga:
1. Fa'ailoga a Leibniz: \( \frac{dy}{dx} \)
2. Fa'ailoga Lagrange: \( f'(x) \)
3. Fa'ailoga a Newton: \( y' \)
4. Fa'ailoga Euler: \( Df(x) \)
E tofu lava faʻailoga ma ona faʻaoga patino ma ona tulaga e masani ona faʻaaogaina ai.
Tulafono Fa'avae i le Fa'avasegaga
Tulafono o le Fa'aopoopoina ma le To'esea
Afai o le \( f(x) \) ma le \( g(x) \) o ni galuega faatino eseese se lua, ona:
\[ \frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x) \]
Tulafono o le Fa'atelega
Mo galuega faatino e lua \( u(x) \) ma le \( v(x) \):
\[ \frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) \]
Tulafono o le Vaevaega
Afai o galuega faatino e lua o le \( u(x) \) ma le \( v(x) \) ma le \( v(x) \neq 0 \):
\[ \frac{d}{dx} \left[ \frac{u(x)}{v(x)} \right] = \frac{u'(x) \cdot v(x) – u(x) \cdot v'(x)}{[v(x)]^2} \]
Tulafono o le Filifiliga
Mo le tuufaatasiga o galuega faatino e lua \( f(u) \) ma le \( u(g) \):
\[ \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) \]
Faʻataʻitaʻiga o le Faʻaaogaina
Mea e Maua mai i Galuega Fa'atino a le Polynomial
Faapea \( f(x) = 3x^3 – 5x^2 + 2x – 1 \). Ina ia maua le derivative o lenei galuega faatino, tatou te faʻaaogaina tulafono faavae o le differentiation.
\[ f'(x) = \frac{d}{dx} (3x^3) – \frac{d}{dx} (5x^2) + \frac{d}{dx} (2x) – \frac{d}{dx} (1) \]
\[ f'(x) = 9x^2 – 10x + 2 \]
Fa'asologa o Galuega Fa'a-Exponential ma Logarithmic
Afai o le \( f(x) = e^x \), o lona uiga o le derivative o le exponential function e faapea:
\[ f'(x) = e^x \]
Mo le galuega faatino o le logarithm masani \( f(x) = \ln(x) \):
\[ f'(x) = \frac{1}{x} \]
Mea e Maua mai i Galuega Taulima Trigonometric
Mo galuega tauave faavae o le trigonometric:
– Afai o le f(x) = sin(x)), ona f'(x) = cos(x) lea
– Afai \( f(x) = \cos(x) \), ona \( f'(x) = -\sin(x) \)
– Afai \( f(x) = \tan(x) \), ona \( f'(x) = \sec^2(x) \)
Fa'asologa o le Galuega Fa'atasi
Faapea \( f(x) = \sin(2x) \). E mafai ona tatou faʻaaogaina le tulafono o le filifili:
\[ f'(x) = \cos(2x) \cdot \frac{d}{dx}(2x) = \cos(2x) \cdot 2 = 2\cos(2x) \]
Fa'asologa Maualuga
Fa'asologa Lua ma Fa'asologa Mulimuli ane
O le lona lua o le derivative o le derivative o le galuega muamua o le derivative. Afai o le \( y = f(x) \) ona fa'ailoa lea o le derivative lona lua e le \( f”(x) \) po'o le \( \frac{d^2y}{dx^2} \). Ma fa'apena fo'i mo le derivative lona tolu \( f”'(x) \) po'o le \( \frac{d^3y}{dx^3} \).
Faapea \( f(x) = x^4 \):
\[ f'(x) = 4x^3 \]
\[ f”(x) = \frac{d}{dx}(4x^3) = 12x^2 \]
\[ f”'(x) = \frac{d}{dx}(12x^2) = 24x \]
\[ f””(x) = \frac{d}{dx}(24x) = 24 \]
Fa'aaogāga o Mea e Maua Mai i Fa'amatalaga i le Fisiki
I le fisiki, e masani ona faʻaaogaina mea e maua mai ai (derives) e fuafua ai le saoasaoa ma le faʻavavevaveina. Faʻapea o le \( s(t) \) o se galuega tauave o le tulaga e faʻatatau i le taimi \( t \). O le saoasaoa \( v(t) \) o le muamua lea o mea e maua mai ai le tulaga:
\[ v(t) = s'(t) \]
O le fa'avavevaveina \( a(t) \) o le ulua'i fa'asologa o le saoasaoa po'o le lona lua fa'asologa o le tulaga:
\[ a(t) = v'(t) = s”(t) \]
I'uga
O le fa'aliliuga o se galuega faatino o se manatu autū i le calculus ma fa'aoga lautele i vaega eseese. O le malamalama fa'alemafaufau i le fa'aliliuga o le fa'asolosolo o se laina tangent e maua ai ni malamalamaaga taua i meatotino ma amioga a se galuega faatino. O le malamalama ma le mafai ona fa'aoga tulafono o le eseesega e pei o le tulafono o le filifili, le tulafono o oloa, ma le tulafono o le vaevaega e taua tele mo so'o se tasi e su'esu'eina le calculus. E ala i fa'ata'ita'iga faigofie ma fa'aoga i le fisiki, o lenei tusiga e fa'amoemoe e tu'uina atu se malamalamaga atoatoa i le tusiaina o le fa'aliliuga o se galuega faatino.
