Ke kākau ʻana i ka Derivative o kahi Hana
Pendahuluan
I ka makemakika, ʻoiai hoʻi ka calculus, he manaʻo nui ka derivative e pāʻani ana i kahi hana koʻikoʻi i nā ʻano noi like ʻole. Hoʻohana ʻia nā derivatives ʻaʻole wale i ka makemakika theoretical akā i ka ʻepekema, ʻenekinia, hoʻokele waiwai, a me nā ʻano aʻo ʻē aʻe he nui. E kūkākūkā kēia ʻatikala i ka derivative o kahi hana me nā kikoʻī, e uhi ana i kāna mau kumu, nā lula koʻikoʻi, a me nā laʻana noi.
Nā Kumu o nā Derivatives
Ka Wehewehena o nā Derivatives
Hōʻike ka derivative o kahi hana i ka wikiwiki o ka loli o ka hana e pili ana i kāna loli kūʻokoʻa. Ma ke ʻano naʻau, hiki ke wehewehe ʻia ka derivative ma ke ʻano he pali o ka laina tangent e hoʻopā ana i ka pakuhi o ka hana ma kahi kiko.
Inā \( y = f(x) \), a laila ua hōʻike ʻia ka derivative mua o \( f \) e pili ana iā \( x \) e \( f'(x) \) a i ʻole \( \frac{dy}{dx} \). Hāʻawi ʻia ka wehewehe kūhelu o ka derivative e ka palena aʻe:
f'(x) = \lim_{{h \to 0}} \frac{f(x+h) – f(x)}{h} \]
Ka Hoʻopaʻa Inoa Hoʻoili
Aia kekahi mau hōʻailona i hoʻohana pinepine ʻia i ke kākau ʻana i nā derivatives:
1. Ka hōʻailona ʻana o Leibniz: \( \frac{dy}{dx} \)
2. Ka hoʻopaʻa inoa ʻana o Lagrange: \( f'(x) \)
3. Ka hōʻailona ʻana o Newton: \( y' \)
4. Ka hoʻopaʻa inoa ʻana o Euler: \( Df(x) \)
Loaʻa i kēlā me kēia hōʻailona nā hoʻohana kikoʻī a me nā pōʻaiapili kahi e hoʻohana nui ʻia ai lākou.
Nā Rula Kumu ma ka Hoʻokaʻawale ʻana
Nā Rula Hoʻohui a me ka Hoʻemi
Inā he mau hana ʻokoʻa ʻelua ʻo \( f(x) \) a me \( g(x) \), a laila:
\[ \frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x) \]
Nā Rula Hoʻonui
No nā hana ʻelua \( u(x) \) a me \( v(x) \):
\[ \frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) \]
Nā Rula Māhele
Inā he mau hana ʻelua ʻo \( u(x) \) a me \( v(x) \) , a ʻo \( v(x) \neq 0 \):
\[ \frac{d}{dx} \left[ \frac{u(x)}{v(x)} \right] = \frac{u'(x) \cdot v(x) – u(x) \cdot v'(x)}{[v(x)]^2} \]
Kānāwai Kaulahao
No ka haku mele ʻana o nā hana ʻelua \( f(u) \) a me \( u(g) \):
\[ \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) \]
Nā Laʻana o ka Hoʻohana
Nā huaʻōlelo i loaʻa mai nā hana Polynomial
Manaʻo \( f(x) = 3x^3 – 5x^2 + 2x – 1 \). No ka loaʻa ʻana o ka derivative o kēia hana, hoʻopili mākou i nā lula kumu o ka hoʻokaʻawale ʻana.
\[ f'(x) = \frac{d}{dx} (3x^3) – \frac{d}{dx} (5x^2) + \frac{d}{dx} (2x) – \frac{d}{dx} (1) \]
\[ f'(x) = 9x^2 – 10x + 2 \]
Nā Derivatives o nā Hana Exponential a me Logarithmic
Inā \( f(x) = e^x \), a laila ʻo ka derivative o ka hana exponential penei:
\[ f'(x) = e^x \]
No ka hana logarithm kūlohelohe \( f(x) = \ln(x) \):
\[ f'(x) = \frac{1}{x} \]
Nā huaʻōlelo i loaʻa mai nā hana trigonometric
No nā hana trigonometric kumu:
– Inā \( f(x) = \sin(x) \), a laila \( f'(x) = \cos(x) \)
– Inā \( f(x) = \cos(x) \), a laila \( f'(x) = -\sin(x) \)
– Inā \( f(x) = \tan(x) \), a laila \( f'(x) = \sec^2(x) \)
Ka Hana Kumu o ka Hana Hui Pū ʻIa
Manaʻo ʻia \( f(x) = \sin(2x) \). Hiki iā kākou ke hoʻopili i ke kānāwai kaulahao:
\[ f'(x) = \cos(2x) \cdot \frac{d}{dx}(2x) = \cos(2x) \cdot 2 = 2\cos(2x) \]
Nā Derivatives Kiʻekiʻe
Nā Derivatives ʻElua a me nā Hope
ʻO ka lua o ka derivative ka derivative o ka hana derivative mua. Inā \( y = f(x) \) a laila ua hōʻike ʻia ka lua o ka derivative e \( f”(x) \) a i ʻole \( \frac{d^2y}{dx^2} \). A pēlā aku no ke kolu o ka derivative \( f”'(x) \) a i ʻole \( \frac{d^3y}{dx^3} \).
Manaʻo ʻia \( f(x) = x^4 \):
\[ f'(x) = 4x^3 \]
\[ f”(x) = \frac{d}{dx}(4x^3) = 12x^2 \]
\[ f”'(x) = \frac{d}{dx}(12x^2) = 24x \]
\[ f””(x) = \frac{d}{dx}(24x) = 24 \]
Nā Hoʻohana o nā Derivatives i ka Physics
I loko o ka physics, hoʻohana pinepine ʻia nā derivatives e hoʻoholo i ka wikiwiki a me ka wikiwiki. Manaʻo ʻia ʻo \( s(t) \) he hana o ke kūlana e pili ana i ka manawa \( t \). ʻO ka wikiwiki \( v(t) \) ka derivative mua o ke kūlana:
\[ v(t) = s'(t) \]
ʻO ka hoʻolalelale \( a(t) \) ka derivative mua o ka wikiwiki a i ʻole ka derivative ʻelua o ke kūlana:
\[ a(t) = v'(t) = s”(t) \]
Ka hopena
ʻO ka derivative o kahi hana he manaʻo nui ia i ka calculus me nā noi ākea ma nā ʻano like ʻole. ʻO ka ʻike maʻalahi o ka derivative e like me ke kiʻekiʻe o kahi laina tangent e hāʻawi i nā ʻike koʻikoʻi i nā waiwai a me ke ʻano o kahi hana. ʻO ka hoʻomaopopo ʻana a me ka hiki ke hoʻopili i nā lula o ka hoʻokaʻawale ʻana e like me ke kānāwai kaulahao, ke kānāwai huahana, a me ke kānāwai mahele he mea nui ia no kekahi e aʻo ana i ka calculus. Ma o nā hiʻohiʻona maʻalahi a me nā noi i ka physics, manaʻolana kēia ʻatikala e hāʻawi i kahi ʻike piha o ke kākau ʻana i ka derivative o kahi hana.
