Fa'ata'ita'iga o fesili ma talanoaga e uiga i meatotino o galuega fa'atino e maua mai ai
O le fa'aliliuga o se galuega faatino o se manatu autū i le calculus e matuā aogā tele mo le iloiloina o amioga a nisi o galuega faatino. I totonu o lenei tusiga, o le a tatou talanoaina ai ni fa'ata'ita'iga o fa'afitauli ma talanoaina ai meatotino o le fa'aliliuga o se galuega faatino.
Fa'atomuaga i Galuega Fa'atino e Maua Mai
O le fa'aliliuga o se galuega faatino \( f \) e fa'aalia o le \( f'(x) \). O le fa'aliliuga muamua o se galuega faatino e tu'uina atu ai le fua faatatau o le suiga o le galuega faatino e fa'atatau i lona fesuia'iga tuto'atasi. O le isi fa'aupuga e masani ona fa'aaogaina o le eseesega. Afai o le \( y = f(x) \), o lona uiga o le fa'aliliuga o le \( f \) e fa'atatau i le \( x \) e fa'apea:
\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) – f(x)}{h} \]
Meatotino o Galuega Fa'atino e Maua Mai
O nisi o meatotino taua o le derivative o se function o:
1. Fa'asologa o le laina: Afai o le \( f(x) \) ma le \( g(x) \) o ni galuega fa'atino e mafai ona fa'avasegaina, ma o le \( c \) o se mea e tumau, ona:
\[
\frac{d}{dx} [cf(x) + g(x)] = c f'(x) + g'(x)
\]
2. Tulafono o le Filifiliga: Mo le galuega faatino tuufaatasi \( g(f(x)) \):
\[
\frac{d}{dx} g(f(x)) = g'(f(x)) \cdot f'(x)
\]
3. Oloa: Mo galuega faatino \( u(x) \) ma le \( v(x) \):
\[
\frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]
4. Quotient: Mo galuega faatino \( u(x) \) ma le \( v(x) \) lea \( v(x) \neq 0 \):
\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) – u(x)v'(x)}{(v(x))^2}
\]
Fesili Fa'ata'ita'i ma Talanoaga
Faʻataʻitaʻiga 1: Fuafuaina o le Derivative o se Galuega Faigofie
Faapea \( f(x) = 3x^2 + 5x – 4 \). Fuafua le fua fa'atatau o le galuega faatino.
Tali:
Matou te faʻaaogaina tulafono faavae o le eseesega.
\[
f(x) = 3x^2 + 5x – 4
\]
Muamua fa'aliliuga:
\[
f'(x) = \frac{d}{dx} (3x^2) + \frac{d}{dx} (5x) – \frac{d}{dx} (4)
\]
Fuafuaina o fa'asologa ta'itasi:
\[
\frac{d}{dx} (3x^2) = 6x
\]
\[
\frac{d}{dx} (5x) = 5
\]
\[
\frac{d}{dx} (4) = 0
\]
O lena la:
\[
f'(x) = 6x + 5
\]
Faataitaiga 2: Faaaogaina o le Tulafono o le Filifiliga
I le tu'uina atu o le galuega faatino \( y = (2x^3 – x^2 + 1)^5 \). Fuafua le fua fa'atatau o le galuega faatino.
Tali:
Faaaoga le tulafono o le filifili (chain rule). Faapea \( u = 2x^3 – x^2 + 1 \), ona mafai lea ona toe tusia le galuega faatino e pei o \( y = u^5 \).
Muamua, saili le derivative o le \( y \) e faatatau i le \( u \):
\[
\frac{dy}{du} = 5u^4
\]
Sosoo ai, saili le derivative o le \( u \) e faatatau i le \( x \):
\[
u = 2x^3 – x^2 + 1
\]
\[
\frac{du}{dx} = 6x^2 – 2x
\]
Tuufaatasia ia mea e lua e maua mai i le tulafono o le filifili:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 5u^4 \cdot (6x^2 – 2x)
\]
Suitulaga toe \( u = 2x^3 – x^2 + 1 \):
\[
\frac{dy}{dx} = 5(2x^3 – x^2 + 1)^4 \cdot (6x^2 – 2x)
\]
Faataitaiga 3: Faaaogaina o Tulafono o Oloa
Ua tuuina mai \( f(x) = x^2 e^x \). Fuafua le derivative o le function.
Tali:
Faaaoga le tulafono o le fua fa'atatau, o lona uiga, afai o le \( u(x) = x^2 \) ma le \( v(x) = e^x \), ona:
\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\]
Muamua, fuafua ia mea e maua mai i le \( u(x) \) ma le \( v(x) \):
\[
u(x) = x^2 \fa'ailoa mai u'(x) = 2x
\]
\[
v(x) = e^x \fa'auigaina v'(x) = e^x
\]
I le faʻaaogaina o tulafono o oloa:
\[
f'(x) = 2x \cdot e^x + x^2 \cdot e^x = e^x (2x + x^2)
\]
Faataitaiga 4: Faaaogaina o le Tulafono o le Quotient
Ua tuuina mai \( f(x) = \frac{x^2 + 1}{x + 2} \). Saili le fua faatatau o le galuega faatino.
Tali:
Faaaoga le tulafono o le quotient, pe afai o le \( u(x) = x^2 + 1 \) ma le \( v(x) = x + 2 \), ona:
\[
f'(x) = \frac{u'(x)v(x) – u(x)v'(x)}{[v(x)]^2}
\]
Muamua, fuafua ia mea e maua mai i le \( u(x) \) ma le \( v(x) \):
\[
u(x) = x^2 + 1 \fa'aalia ai u'(x) = 2x
\]
\[
v(x) = x + 2 \fa'aalia ai le v'(x) = 1
\]
I le faʻaaogaina o le tulafono o le quotient:
\[
f'(x) = \frac{2x(x + 2) – (x^2 + 1)(1)}{(x + 2)^2}
\]
\[
f'(x) = \frac{2x^2 + 4x – x^2 – 1}{(x + 2)^2}
\]
\[
f'(x) = \frac{x^2 + 4x – 1}{(x + 2)^2}
\]
I'uga
I le calculus, o le malamalama i le manatu autū o mea e maua mai ai ma o latou meatotino e taua tele mo le foia o faʻafitauli eseese o le matematika. O lenei tusiga o loʻo aoteleina ai ni metotia eseese mo le mauaina o galuega faatino e ala i le faʻaalia o le faʻaaogaina o tulafono autū e pei o le linearity, chains, products, ma quotients e ala i ni faʻataʻitaʻiga ma talanoaga auiliili. O le malamalama ma le faʻataʻitaʻi soo o mea e maua mai ai, e mafai ona tatou sili atu ona tomai i le iloiloina o suiga i galuega faatino i tulaga eseese.
