Nā Hoʻohana o nā Derivatives ma nā ʻAno ʻEpekema like ʻole
He manaʻo nui ka derivative i ka calculus, i hoʻolauna ʻia e Sir Isaac Newton lāua ʻo Gottfried Wilhelm Leibniz i ka hopena o ke kenekulia 17. I ka makemakika, hōʻike ka derivative i ka wikiwiki o ka loli o kahi hana e pili ana i kekahi o kāna mau loli. ʻAʻole wale kēia manaʻo he mea nui i ka makemakika ponoʻī akā he mau noi ākea hoʻi i nā ʻano ʻepekema like ʻole. E kūkākūkā kēia ʻatikala i nā noi like ʻole o nā derivatives i nā ʻano aʻo like ʻole, mai ka physics a me ka hoʻokele waiwai a hiki i ka biology a me ka ʻenekinia a hiki i ka ʻepekema kamepiula.
1. ʻIke kino
He mea nui loa nā derivatives i ka physics, ʻoi aku hoʻi i ka mechanics classical. ʻO kekahi o nā hiʻohiʻona maʻamau loa ʻo ia ka pilina ma waena o ke kūlana, ka wikiwiki, a me ka wikiwiki. Inā ʻo \( s(t) \) ke kūlana o kahi mea ma ke ʻano he hana o ka manawa, a laila:
– ʻO ka wikiwiki (\( v(t) \)) ka derivative mua o ke kūlana e pili ana i ka manawa: \( v(t) = \frac{ds(t)}{dt} \).
– ʻO ka hoʻolalelale ʻana (\( a(t) \)) ka derivative mua o ka wikiwiki a i ʻole ka derivative ʻelua o ke kūlana e pili ana i ka manawa: \( a(t) = \frac{dv(t)}{dt} = \frac{d^2s(t)}{dt^2} \).
Eia kekahi, i ka electromagnetism, ʻōlelo ke kānāwai o Faraday no ka induction electromagnetic ʻo ka ikaika electromotive (EMF) i hoʻokomo ʻia i loko o kahi kaapuni ʻo ia ka derivative o ka flux magnetic e pili ana i ka manawa.
2. Hoʻokele waiwai
I loko o ka hoʻokele waiwai, hoʻohana ʻia nā derivatives e kālailai i nā loli i ke kumukūʻai, ka loaʻa kālā, a me nā hana hana. Eia kekahi laʻana:
– ʻO ke kumukūʻai palena (Biaya Marginal) ka derivative o ka hana kumukūʻai holoʻokoʻa e pili ana i ka nui o ka hana, e hōʻike ana pehea e loli ai nā kumukūʻai holoʻokoʻa i ka piʻi ʻana o ka hana: \( MC = \frac{dTC}{dQ} \).
– ʻO ka Loaʻa Kūʻē (Loaʻa Kūʻē) ka hopena o ka hana loaʻa kālā holoʻokoʻa e pili ana i ka nui o ka hana, e wehewehe ana i ka loaʻa kālā hou ke hoʻonui ʻia ke kūʻai aku: \( MR = \frac{dTR}{dQ} \).
ʻO kekahi noi koʻikoʻi i loko o ke kumumanaʻo pono. Hōʻike ka hana pono i ka ʻoluʻolu a i ʻole ka pono i loaʻa mai ka hoʻohana ʻana i nā waiwai a me nā lawelawe. ʻO ka derivative mua o ka hana pono i kapa ʻia ʻo marginal utility, e hōʻike ana i ka ʻoluʻolu hou i loaʻa mai ka hoʻopau ʻana i kahi ʻāpana hou o kahi waiwai a lawelawe paha.
3. ʻIkeolaola
I loko o ka biology, hoʻohana ʻia nā derivatives e hoʻohālike i nā kaʻina hana like ʻole. No ka laʻana, i loko o ka ecology, hoʻohana pinepine nā hiʻohiʻona ulu heluna kanaka i nā derivatives e wehewehe i ka nui o ka ulu ʻana o kahi heluna kanaka. ʻO nā hiʻohiʻona ulu exponential a me logistic ʻelua mau hiʻohiʻona maʻamau:
– Ke kumu hoʻohālike ulu exponential: \( \frac{dN}{dt} = rN \), kahi ʻo \( N \) ka nui o ka heluna kanaka a ʻo \( r \) ka helu ulu kūloko.
– Ke kumu hoʻohālike ulu logistic: \( \frac{dN}{dt} = rN \left( 1 – \frac{N}{K} \right) \), kahi ʻo \( K \) ka mana kaiapuni a ʻo \( \left( 1 – \frac{N}{K} \right) \) he kumu hoʻemi e hōʻemi ana i ka wikiwiki o ka ulu ʻana i ka wā e hoʻokokoke ʻia ai ka mana kaiapuni.
I ke ʻano o ka physiology, hoʻohana ʻia nā derivatives e hoʻohālike i nā kaʻina hana physiological e like me ke kahe koko a me ka hoʻoili ʻana o nā hōʻailona nerve. No ka laʻana, hoʻohana ke kānāwai o Darcy no ke kahe koko i loko o nā kīʻaha koko i nā derivatives e wehewehe i nā loli o ke koko ma ke kīʻaha koko.
4. ʻAno hana
He mea nui loa hoʻi nā Derivatives i nā lālā like ʻole o ka ʻenekinia. I ka ʻenekinia kīwila a me nā mechanics, hoʻohana ʻia nā derivatives i ka loiloi kūkulu a me nā mechanics fluid. Eia kekahi laʻana:
– I ka loiloi hoʻonohonoho, hoʻohana ʻia ke ʻano hana finite element e hoʻoholo ai i nā deformations a me nā stress i nā hale. Hāʻawi ka derivative o ka hana displacement i ke kaumaha, a hāʻawi ka derivative o ke kaumaha i ke kaumaha.
– I loko o ka mechanics fluid, wehewehe nā kaulike Navier-Stokes i ke kahe ʻana o ka wai. ʻO kēia mau kaulike he mau kaulike ʻokoʻa e pili ana i nā derivatives hapa o ka wikiwiki o ka wai e pili ana i ka manawa a me ka hakahaka.
I ke ʻenekinia uila, hoʻohana ʻia nā derivatives i ka nānā ʻana i nā kaapuni uila. Hoʻopili pinepine nā kānāwai o Kirchhoff a me ke kumumanaʻo pūnaewele i nā derivatives e wehewehe i ka pilina ma waena o ke au, ka voltage, a me ka inductance i nā kaapuni uila.
5. ʻEpekema Kamepiula
I ka ʻepekema kamepiula, hoʻohana ʻia nā derivatives i ka optimization a me ke aʻo ʻana i ka mīkini. ʻO kekahi o nā noi nui i nā algorithms optimization e like me ka gradient descent. Hoʻohana ʻia kēia algorithm e hōʻemi i kahi hana kumukūʻai i ke kaʻina hana aʻo o kahi kumu hoʻohālike aʻo mīkini:
– ʻO ka gradient ka vector derivative mua o ka hana kumukūʻai e pili ana i nā palena hoʻohālike, e hōʻike ana i ke kuhikuhi o ka loli nui loa.
– ʻO ka gradient descent kahi hana iterative e hoʻohana ana i nā gradients e hōʻano hou i nā palena hoʻohālike i ka palena iki o ka hana kumukūʻai.
Eia kekahi, i nā kiʻi kamepiula, hoʻohana ʻia nā derivatives e hoʻoponopono i ka kukui a me nā aka. Hoʻohana ke kānāwai o Lambert a me ke kumu hoʻohālike kukui Phong i nā derivatives e helu i ka ikaika o ka mālamalama i hōʻike ʻia mai kahi ʻili i hōʻike ʻia i kahi kumu kukui.
6. Kemika
I loko o ke kemika, hoʻohana ʻia nā derivatives i loko o ke kinetics reaction e wehewehe i ka wikiwiki o kahi reaction kemika. Hōʻike pinepine ʻia ka wikiwiki o ka reaction ma ke ʻano he derivative o ka nui o kahi reactant a huahana paha e pili ana i ka manawa. No ka laʻana, no ka reaction mua:
\[ \text{Nā Mea Hana} \rightarrow \text{Nā Huahana} \]
Hiki ke hōʻike ʻia ka wikiwiki o ka hopena \( r(t) \) penei:
\[ r(t) = – \frac{d[\text{Mea Hana}]}{dt} = \frac{d[\text{Huahana}]}{dt} \]
Eia kekahi, hoʻohana ʻia nā derivatives i ka thermodynamics e kālailai i nā loli ikehu i loko o kahi ʻōnaehana. No ka laʻana, ʻo ka Gibbs free energy (G) kahi hana thermodynamic i hoʻohana pinepine ʻia e wānana i ke kuhikuhi o nā hopena kemika, a ʻo ka derivative mua o G e pili ana i nā kekelē o ke kūʻokoʻa o kahi ʻōnaehana e hāʻawi i ka ʻike e pili ana i ke kūlana o ke kaulike thermodynamic.
Ka hopena
Mai ka wehewehe ʻana ma luna, ua maopopo he nui nā noi o ke kumumanaʻo o nā derivatives ma nā ʻano ʻepekema like ʻole. I ka physics, wehewehe nā derivatives i ka pilina kumu ma waena o ke kūlana, ka wikiwiki, a me ka wikiwiki. I ka hoʻokele waiwai, hoʻohana ʻia nā derivatives e kālailai i nā kumukūʻai marginal a me nā loaʻa kālā. I ka biology, kōkua nā derivatives i ke kumu hoʻohālike o ka ulu ʻana o ka heluna kanaka a me nā kaʻina hana physiological. I ka ʻenekinia, he mea nui nā derivatives i ka loiloi kūkulu a me nā mechanics fluid. Hoʻohana ka ʻepekema kamepiula i nā derivatives i nā algorithms optimization a me ke aʻo mīkini. I ka kemika, hoʻohana ʻia nā derivatives i nā kinetics reaction a me thermodynamics. No laila, he mea nui ka hoʻomaopopo ʻana a me ka hoʻokele ʻana i ke kumumanaʻo o nā derivatives no nā kānaka ʻepekema a me nā ʻenekinia ma nā ʻano aʻo like ʻole.
