Nā Hana Algebraic: He Alakaʻi Piha
ʻO ke kumu o kahi hana he manaʻo nui ia i ka calculus a me ka makemakika ma ke ʻano laulā. ʻAʻole pili kēia manaʻo i ke kumumanaʻo wale nō akā he mau noi kūpono hoʻi i nā ʻano like ʻole, me ke kino, ka ʻenekinia, ka hoʻokele waiwai, a me ka ʻepekema kamepiula. E kūkākūkā kēia ʻatikala i ke kumu algebra o kahi hana, mai kona wehewehe kumu a hiki i kāna mau noi paʻakikī.
Ka Wehewehena o nā Derivatives
I ka makemakika, 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 manaʻo ʻia ka derivative e like me ke kiʻekiʻe o ka laina tangent i kahi piʻo ma kahi kiko i hāʻawi ʻia. Inā \( y = f(x) \), a laila ua hōʻike ʻia ka derivative o ka hana ʻo \( f'(x) \) a i ʻole \( \frac{dy}{dx} \).
Palena Hoʻokokoke
Hoʻohana ka wehewehe kūhelu o kahi derivative i ke kumumanaʻo o kahi palena. Inā he hana hoʻomau ʻo \( f(x) \), a laila ua wehewehe ʻia ka derivative mua o ka hana penei:
\[
f'(x) = \lim_{{h \to 0}} \frac{f(x+h) – f(x)}{h}
\]
Maanei, ʻo \( h \) kahi loli liʻiliʻi ma \( x \). ʻO kēia palena, inā loaʻa, hāʻawi i ka helu loli maikaʻi loa a i ʻole ka pali o \( f(x) \) ma ke kiko \( x \).
Nā Rula Kumu ma ka Hoʻokaʻawale ʻana
1. Kānāwai Paʻa:
Inā he kūpaʻa ʻo \( c \) a ʻo \( f(x) = c \), a laila:
\[
f'(x) = 0
\]
2. Nā Rula Kūlana:
Inā \( f(x) = x^n \) no kekahi helu maoli \( n \), a laila:
\[
f'(x) = nx^{n-1}
\]
3. Kānāwai Paʻa Pālua:
Inā \( f(x) = cg(x) \) no kekahi hana \( g(x) \) a me ke kūpaʻa \( c \), a laila:
\[
(cf(x))' = c f'(x)
\]
4. Nā Rula Hoʻohui:
Inā he mau hana ʻokoʻa ʻelua ʻo \( f(x) \) a me \( g(x) \), a laila:
\[
(f(x) + g(x))' = f'(x) + g'(x)
\]
5. Nā Rula Hoʻonui:
Inā he mau hana ʻokoʻa ʻelua ʻo \( f(x) \) a me \( g(x) \), a laila:
\[
(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)
\]
6. Nā Rula Māhele:
No nā hana ʻelua \( f(x) \) a me \( g(x) \) i hiki ke hoʻokaʻawale ʻia e \( g(x) \neq 0 \), a laila:
\[
\left( \frac{f(x)}{g(x)} \right)' = \frac{f'(x)g(x) – f(x)g'(x)}{g(x)^2}
\]
7. Kānāwai Kaulahao:
Inā ʻo \( y = f(u) \) a me \( u = g(x) \), a laila:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
\]
Nā Laʻana Noi
Laʻana 1: Manaʻo ʻia \( f(x) = 4x^3 – 2x + 7 \). A laila hiki ke helu ʻia ka derivative o \( f(x) \) penei:
\[
f'(x) = 12x^2 – 2
\]
Maanei, hoʻopili mākou i ka lula mana a me ka lula mau pālua.
_Laʻana 2:_ E noʻonoʻo \( g(x) = (2x^2 – 3x)(x^3 + 1) \). No ka loaʻa ʻana o \( g'(x) \), hoʻohana mākou i ke kānāwai hoʻonui:
\[
g'(x) = (2x^2 – 3x)'(x^3 + 1) + (2x^2 – 3x)(x^3 + 1)'
\]
\[
= (4x – 3)(x^3 + 1) + (2x^2 – 3x)(3x^2)
\]
\[
= 4x(x^3 + 1) – 3(x^3 + 1) + 6x^4 – 9x^3
\]
\[
= 4x^4 + 4x – 3x^3 – 3 + 6x^4 – 9x^3
\]
\[
= 10x^4 – 12x^3 + 4x – 3
\]
Nā Hoʻohana Ola Maoli o nā Derivatives
1. ʻIke kino:
Hoʻohana pinepine ka ʻepekema kino i nā derivatives e hoʻomaopopo i nā manaʻo o ka wikiwiki a me ka wikiwiki. No ka laʻana, inā ʻo \( s(t) \) ke kūlana o kahi mea ma ke ʻano he hana o ka manawa \( t \), a laila ʻo ka wikiwiki \( v(t) \) ka derivative mua o ke kūlana \( s(t) \), a ʻo ka wikiwiki \( a(t) \) ka derivative o ka wikiwiki.
2. Hoʻokele waiwai:
I ka hoʻokele waiwai, hoʻohana ʻia nā derivatives e ʻike i ka helu o ka loli marginal. ʻO nā hiʻohiʻona o nā noi e komo pū me ke kumukūʻai marginal, e wehewehe ana pehea e loli ai nā kumukūʻai holoʻokoʻa me ka hana ʻana o hoʻokahi ʻāpana hou aʻe.
3. ʻAno hana:
I loko o ka ʻenekinia, hoʻohana ʻia nā derivatives no ka nānā ʻana i ke kūpaʻa a me ka kaohi ʻōnaehana. No ka laʻana, i loko o ka mechanics structural, hoʻohana ʻia nā derivatives e hoʻoholo i nā koʻikoʻi a me nā strains i nā mea.
4. Nā Kiʻikuhi a me nā Piʻo:
Hoʻohana ʻia nā Derivatives e ʻike i nā kiko kiʻekiʻe a me nā palena iki ma ke kaʻe o kahi hana, he mea nui ia i ka optimization.
Ka hopena
ʻO ke akamai i ke kumumanaʻo o ka derivative o nā hana algebraic he mea nui ia no ka hoʻomaopopo ʻana i nā hanana makemakika like ʻole a me kā lākou mau noi ola maoli. Ma ka hoʻohana ʻana i nā lula kumu o ka hoʻokaʻawale ʻana, hiki iā mākou ke loaʻa maʻalahi i nā derivatives o nā hana like ʻole a hoʻopili iā lākou e hoʻoponopono i nā pilikia o ka honua maoli ma nā ʻano like ʻole. Me ka manaʻolana, hāʻawi kēia ʻatikala i kahi ʻike piha o nā derivatives o nā hana algebraic.
Kuhikuhi
No ka hoʻoikaika ʻana i kou ʻike i nā derivatives, paipai nui mākou e heluhelu i kahi puke aʻo calculus e like me "Calculus" na James Stewart a i ʻole "Advanced Calculus" na Michael Spivak. Eia kekahi, hiki i nā kumuwaiwai pūnaewele like ʻole a me nā aʻo wikiō ke kōkua nui.
