Te Tuhi i te Pānga o tētahi Mahi
Pendahuluan
I roto i te pāngarau, inā koa te tātaitai, he ariā taketake te taupū e whai wāhi nui ana ki te whānuitanga o ngā tono. E whakamahia ana ngā taupū ehara i te mea i roto i te pāngarau ariā anake engari i roto hoki i te pūtaiao, te miihini, te ōhanga, me te maha atu o ngā kaupapa ako. Ka matapakihia e tēnei tuhinga te taupū o tētahi mahi me te taipitopito, e kapi ana i ōna kaupapa taketake, ngā ture nui, me ngā tauira tono.
Ngā Kaupapa Taketake o ngā Hua Whakaputa
Te Whakamāramatanga o ngā Hua Whakaputa
Ko te tātaitanga o tētahi mahi e whakaahua ana i te tere o te huringa o te mahi e pā ana ki tōna taurangi motuhake. Mā te whakaaro whānui, ka taea te tautuhi i te tātaitanga hei te pikinga o te rārangi pātapa e pā ana ki te kauwhata o te mahi i tētahi pūwāhi.
Mena ko \( y = f(x) \), ko te tuatahi o ngā pānga o \( f \) e pā ana ki \( x \) ka tohua ko \( f'(x) \) ko \( \frac{dy}{dx} \ rānei). Ko te whakamāramatanga ōkawa o te pānga ka hoatu e te rohe e whai ake nei:
\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) – f(x)}{h} \]
Tohu Whakaputa
He maha ngā momo tohu e whakamahia whānuitia ana hei tuhi i ngā kupu whakataki:
1. Te tuhi a Leibniz: \( \frac{dy}{dx} \)
2. Te tuhipoka Lagrange: \( f'(x) \)
3. Te tuhi a Newton: \( y' \)
4. Te tuhipoka a Euler: \( Df(x) \)
He whakamahinga motuhake tō ia tuhipoka, me ōna horopaki e whakamahia nuitia ai.
Ngā Ture Taketake mō te Whakarerekētanga
Ngā Ture Tāpiri me te Tango
Mena he mahi rerekē e rua a \( f(x) \) me \( g(x) \), kāti:
\[ \frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x) \]
Ngā Ture Whakarea
Mō ngā mahi e rua \( u(x) \) me \( v(x) \):
\[ \frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) \]
Ngā Ture Wehewehenga
Mena he mahi e rua a \( u(x) \) me \( v(x) \) , ā, ko \( v(x) \neq 0 \):
\[ \frac{d}{dx} \left[ \frac{u(x)}{v(x)} \right] = \frac{u'(x) \cdot v(x) – u(x) \cdot v'(x)}{[v(x)]^2} \]
Ture mekameka
Mō te hanganga o ngā mahi e rua \( f(u) \) me \( u(g) \):
\[ \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) \]
Ngā Tauira o te Whakamahinga
Ngā Pānga o ngā Mahi Pūrau
Me kī \( f(x) = 3x^3 – 5x^2 + 2x – 1 \). Hei kimi i te pānga o tēnei mahi, ka whakamahia e mātou ngā ture taketake o te wehewehe.
\[ f'(x) = \frac{d}{dx} (3x^3) – \frac{d}{dx} (5x^2) + \frac{d}{dx} (2x) – \frac{d}{dx} (1) \]
\[ f'(x) = 9x^2 – 10x + 2 \]
Ngā Pānga o ngā Mahi Taupū me ngā Mahi Logarithmic
Mena ko te \( f(x) = e^x \), ko te pānga o te mahi taupū ko:
\[ f'(x) = e^x \]
Mō te mahi logarithm tūturu \( f(x) = \ln(x) \):
\[ f'(x) = \frac{1}{x} \]
Ngā Huanga o ngā Mahi Pāngatoru
Mō ngā mahi pākoki taketake:
– Mēnā ko te f(x) = te sin(x), kāti ko te f'(x) = te cos(x)
– Mēnā ko te f(x) = cos(x) te uara, kāti ko te f'(x) = -sin(x) te uara.
– Mēnā ko te f(x) = te tan(x), kāti ko te f'(x) = te sec^2(x)
Te Pūtake o te Mahi Whakakotahi
Me kī \( f(x) = \sin(2x) \). Ka taea e tātou te whakamahi i te ture mekameka:
\[ f'(x) = \cos(2x) \cdot \frac{d}{dx}(2x) = \cos(2x) \cdot 2 = 2\cos(2x) \]
Ngā Hua Whakaputa Matatau
Ngā Hua Tuarua me ngā Hua Whai Muri Mai
Ko te tuarua o ngā tātaitanga ko te tātaitanga o te mahi tātaitanga tuatahi. Mēnā ko \( y = f(x) \) ka tohua te tuarua o ngā tātaitanga e \( f”(x) \) me \( \frac{d^2y}{dx^2} \). Pērā tonu mō te tuatoru o ngā tātaitanga \( f”'(x) \) me \( \frac{d^3y}{dx^3} \).
Me kī \( f(x) = x^4 \):
\[ f'(x) = 4x^3 \]
\[ f”(x) = \frac{d}{dx}(4x^3) = 12x^2 \]
\[ f”'(x) = \frac{d}{dx}(12x^2) = 24x \]
\[ f””(x) = \frac{d}{dx}(24x) = 24 \]
Ngā Whakamahinga o ngā Hua Whakapūtanga i roto i te Ahupūngao
I roto i te ahupūngao, he maha ngā wā ka whakamahia ngā tātaitanga hei whakatau i te tere me te whakaterenga. Me kī ko te \( s(t) \) he pānga o te tūranga e pā ana ki te wā \( t \). Ko te tere \( v(t) \) te tātaitanga tuatahi o te tūranga:
\[ v(t) = s'(t) \]
Ko te whakaterenga \( a(t) \) te tuatahi o ngā pānga o te tere, te tuarua rānei o ngā pānga o te tūnga:
\[ a(t) = v'(t) = s”(t) \]
Whakamutunga
Ko te pānga o tētahi mahi he ariā taketake i roto i te tātaitai me ngā tono whānui puta noa i ngā mara maha. Mā te mārama ā-ringa ki te pānga hei te pikinga o tētahi rārangi pātata ka homai he māramatanga nui ki ngā āhuatanga me te whanonga o tētahi mahi. He mea nui te mārama me te āhei ki te whakamahi i ngā ture wehewehe pēnei i te ture mekameka, te ture hua, me te ture wehewehe mō te hunga e ako ana i te tātaitai. Mā roto i ngā tauira me ngā tono ngāwari i roto i te ahupūngao, ko te tumanako o tēnei tuhinga ka whakarato i tētahi māramatanga whānui mō te tuhi i te pānga o tētahi mahi.
