Nā nīnau hoʻohālike a me ke kūkākūkā ʻana o nā huaʻōlelo hana
He manaʻo nui ka derivative i ka calculus e pāʻani ana i kahi hana koʻikoʻi i nā noi like ʻole o ka makemakika, physics, engineering, a me nā ʻepekema ʻē aʻe. Ma kēia ʻatikala, e kūkākūkā mākou i kekahi mau laʻana o nā derivatives a me kā lākou mau hoʻonā. ʻO ka hoʻomaopopo ʻana i ke kumumanaʻo o ka derivative e maʻalahi ai ka hoʻopili ʻana iā ia i nā pilikia like ʻole.
Ka Hoʻomaopopo Kumu o nā Derivatives
Hōʻike ka derivative o kahi hana i ka wikiwiki o ka loli o ka hana e pili ana i ka loli kūʻokoʻa. Ma ke ʻano maʻamau, ʻo ka derivative o ka hana \( f(x) \) ma ke kiko \( x \) ʻo ia ka pali o ka laina tangent i ka piʻo \( f \) ma ke kiko \( x \). ʻO ka hōʻailona maʻamau i hoʻohana ʻia no ka derivative ʻo \( f'(x) \) a i ʻole \( \frac{df}{dx} \).
Nā Rula Kumu o nā Derivatives
No ka hoʻoponopono ʻana i nā pilikia derivative, pono mākou e ʻike i kekahi mau lula kumu o nā derivatives:
1. Ka Hoʻoili Kūpaʻa: Inā he kūpaʻa ʻo \( c \), a laila ʻo ka hoʻoili o \( c \) he ʻole.
\[
\frac{d}{dx}(c) = 0
\]
2. Ka hopena o ka hana laina: Inā ʻo \( f(x) = mx + b \), kahi ʻo \( m \) a me \( b \) he mau mea mau, a laila:
\[
f'(x) = m
\]
3. Mana Mana: Inā ʻo \( f(x) = x^n \), kahi ʻo \( n \) he helu maoli, a laila:
\[
f'(x) = nx^{n-1}
\]
4. Rula Huina: Inā \( f(x) = g(x) + h(x) \), a laila:
\[
f'(x) = g'(x) + h'(x)
\]
5. Rula Hoʻonui: Inā \( f(x) = g(x) \cdot h(x) \), a laila:
\[
f'(x) = g'(x)h(x) + g(x)h'(x)
\]
6. Rula Māhele: Inā \( f(x) = \frac{g(x)}{h(x)} \), a laila:
\[
f'(x) = \frac{g'(x)h(x) – g(x)h'(x)}{h(x)^2}
\]
7. Kānāwai Kaulahao: Inā \( f(x) = g(h(x)) \), a laila:
\[
f'(x) = g'(h(x)) \cdot h'(x)
\]
Nā Nīnau Laʻana a me ke Kūkākūkā
Laʻana Nīnau 1
Nīnau: E hoʻoholo i ka derivative o \( f(x) = 3x^2 + 2x + 1 \).
Kūkākūkā:
No ka hoʻoholo ʻana i ka derivative o ka hana, e hoʻohana mākou i ka lula mana a me ka lula huina.
\[
f(x) = 3x^2 + 2x + 1
\]
ʻO nā mea i loaʻa mai:
\[
f'(x) = \frac{d}{dx}(3x^2) + \frac{d}{dx}(2x) + \frac{d}{dx}(1)
\]
Ma muli o ke kānāwai kūlana:
\[
\frac{d}{dx}(3x^2) = 3 \cdot 2x^{2-1} = 6x
\]
\[
\frac{d}{dx}(2x) = 2 \cdot 1x^{1-1} = 2
\]
\[
\frac{d}{dx}(1) = 0
\]
No laila, ʻo ke kumu o ka hana \( f \) penei:
\[
f'(x) = 6x + 2
\]
Laʻana Nīnau 2
Nīnau: E hoʻoholo i ka derivative o ka hana \( g(x) = (2x^3 – x)(x^2 + 3) \).
Kūkākūkā:
No ka hoʻoponopono ʻana i kēia pilikia, e hoʻohana mākou i ke kānāwai hoʻonui.
\[
g(x) = (2x^3 – x)(x^2 + 3)
\]
No laila, ʻo ka derivative o \( g(x) \) penei:
\[
g'(x) = (2x^3 – x)'(x^2 + 3) + (2x^3 – x)(x^2 + 3)'
\]
ʻO ka mea mua, hoʻoholo mākou i ka derivative o kēlā me kēia hana:
\[
(2x^3 – x)' = 6x^2 – 1
\]
\[
(x^2 + 3)' = 2x
\]
A laila hoʻololi mākou iā ia i loko o ke ʻano:
\[
g'(x) = (6x^2 – 1)(x^2 + 3) + (2x^3 – x)(2x)
\]
A laila, hāʻawi mākou:
\[
g'(x) = 6x^2 \cdot x^2 + 6x^2 \cdot 3 – 1 \cdot x^2 – 1 \cdot 3 + 2x^3 \cdot 2x – x \cdot 2x
\]
\[
g'(x) = 6x^4 + 18x^2 – x^2 – 3 + 4x^4 – 2x^2
\]
ʻO ka hope loa, loaʻa iā mākou:
\[
g'(x) = 10x^4 + 15x^2 – 3
\]
Laʻana Nīnau 3
Nīnau: E huli i ka derivative o \( h(x) = \frac{x^2 + 1}{x – 1} \).
Kūkākūkā:
No ka hoʻoponopono ʻana i kēia pilikia, e hoʻohana mākou i ke kānāwai mahele.
\[
h(x) = \frac{x^2 + 1}{x – 1}
\]
No laila, ʻo ka derivative o \( h(x) \) penei:
\[
h'(x) = \frac{(x^2 + 1)'(x – 1) – (x^2 + 1)(x – 1)'}{(x – 1)^2}
\]
ʻO ka mea mua, hoʻoholo mākou i ka derivative o kēlā me kēia hana:
\[
(x^2 + 1)' = 2x
\]
\[
(x – 1)' = 1
\]
A laila hoʻololi mākou iā ia i loko o ke ʻano:
\[
h'(x) = \frac{2x(x – 1) – (x^2 + 1)(1)}{(x – 1)^2}
\]
A laila, hāʻawi mākou:
\[
h'(x) = \frac{2x^2 – 2x – x^2 – 1}{(x – 1)^2}
\]
A laila hoʻomaʻalahi mākou:
\[
h'(x) = \frac{x^2 – 2x – 1}{(x – 1)^2}
\]
Ka hopena
ʻO ka derivative o kahi hana he manaʻo nui ia i ka calculus e hāʻawi ana i ka ʻike e pili ana i ka wikiwiki o ka loli o ka waiwai o kahi hana e pili ana i kāna loli kūʻokoʻa. Ma ka hoʻomaopopo ʻana i nā lula kumu o ka derivation, e like me ka derivative o kahi mau, nā hana linear, ka lula mana, ka huina, ka hoʻonui ʻana, a me ka mahele ʻana, a me ke kānāwai kaulahao, hiki iā mākou ke hoʻoponopono i nā pilikia derivative like ʻole.
ʻO nā pilikia hoʻohālike i kūkākūkā ʻia ma luna nei he hana mua maikaʻi ia i ka hoʻomaopopo ʻana pehea e hoʻopili ai i ke kumumanaʻo o nā derivatives. I ka hana maoli, e hoʻomaikaʻi hou ʻia nā mākau i ka helu ʻana i nā derivatives ma ka hana ʻana me nā ʻano pilikia like ʻole a me nā ʻano like ʻole o nā hana. Manaʻolana, ua kōkua kēia ʻatikala i ka hoʻomaopopo ʻana a me ka hoʻopaʻa ʻana i ke kumumanaʻo o ka derivative o kahi hana.
