Nā nīnau hoʻohālike a me ke kūkākūkā ʻana o ke Chain Rule ma Derivatives
ʻO ke kānāwai kaulahao kekahi o nā manaʻo koʻikoʻi loa i ka calculus differential, i hoʻohana ʻia e helu i ka derivative o kahi hana i haku ʻia me ʻelua a ʻoi aku paha nā hana. Ma kēia ʻatikala, e kūkākūkā mākou i ke kumumanaʻo kumu o ke kānāwai kaulahao, pehea e hoʻohana ai, a me nā hiʻohiʻona o kona hoʻohana ʻana i nā pilikia derivative e kū pinepine ana ma ke kula kiʻekiʻe a me ke kulanui.
1. Hoʻolauna i ke Kānāwai Kaulahao
Ma mua o ko kākou komo ʻana i ka pilikia hoʻohālike, e hoʻomaopopo mua kākou i ke ʻano o ke kānāwai kaulahao. ʻŌlelo ke kānāwai kaulahao inā loaʻa iā kākou ʻelua mau hana like ʻole \( f \) a me \( g \), a makemake mākou e ʻimi i ka derivative o ka haku mele ʻana o nā hana \( h = f(g(x)) \), a laila ʻo ka derivative o \( h \) penei:
\[ h'(x) = f'(g(x)) \cdot g'(x) \]
Ma nā huaʻōlelo maʻalahi, helu mākou i ka derivative o ka hana waho ma g(x), a laila hoʻonui i ka hopena me ka derivative o ka hana kūloko \( g(x) \).
2. Ke Hoʻomaopopo nei i ka Hana o ka Hoʻokumu ʻana
Ma mua o ko kākou komo ʻana i nā pilikia hoʻohālike, he mea nui e hoʻomaopopo i nā hana haku mele. ʻO ka hana haku mele kahi hana i loaʻa ma ka hoʻokomo ʻana i hoʻokahi hana i loko o kekahi. No ka laʻana, inā loaʻa iā kākou \( f(x) = \sin(x) \) a me \( g(x) = x^2 \), a laila ʻo ka haku mele o nā hana ʻelua e like me \( h(x) = f(g(x)) = \sin(x^2) \).
I nā hana haku mele, manaʻo pinepine mākou iā \( g(x) \) ʻo ia ka "hana kūloko" a ʻo \( f(x) \) ʻo ia ka "hana kūwaho". Ma kēia laʻana, ʻo ka hana kūloko ʻo \( x^2 \) a ʻo ka hana kūwaho ʻo sine.
3. Nā nīnau hoʻohālike a me ke kūkākūkā ʻana
E nānā kākou i kekahi mau pilikia hoʻohālike e hoʻohana ana i ke kānāwai kaulahao e hoʻoponopono iā lākou.
Laʻana 1:
Hāʻawi ʻia i ka hana \( y = \cos(3x^2) \), e huli i ka derivative mua o y e pili ana iā x.
Kūkākūkā:
ʻO ka mea mua, ʻike mākou i nā hana o loko a me waho. Maanei, ʻo ka hana o loko ʻo \( g(x) = 3x^2 \) a ʻo ka hana o waho ʻo \( f(g) = \cos(g) \).
Ua ʻike mākou:
1. \( g'(x) = 6x \)
2. \( f'(g) = -\sin(g) \)
Ma ke kānāwai kaulahao, loaʻa iā mākou:
\[ y' = f'(g(x)) \cdot g'(x) = -\sin(3x^2) \cdot 6x \]
No laila, ʻo ke kumu o \( y = \cos(3x^2) \) penei:
\[ y' = -6x \sin(3x^2) \]
Laʻana 2:
E huli i ka derivative mua o \( h(x) = e^{5x^3 + 2x} \).
Kūkākūkā:
Eia ka hana o loko ʻo \( g(x) = 5x^3 + 2x \) a ʻo ka hana o waho ʻo \( f(g) = e^g \).
Ua ʻike mākou:
1. \( g'(x) = 15x^2 + 2 \)
2. \( f'(g) = e^g \)
Ma ke kānāwai kaulahao, loaʻa iā mākou:
\[ h'(x) = f'(g(x)) \cdot g'(x) = e^{5x^3 + 2x} \cdot (15x^2 + 2) \]
No laila, ʻo ka derivative o \( h(x) = e^{5x^3 + 2x} \) penei:
\[ h'(x) = (15x^2 + 2)e^{5x^3 + 2x} \]
Laʻana 3:
E huli i ka derivative mua o \( y = \ln(4x^2 – 5) \).
Kūkākūkā:
ʻO ka hana o loko ʻo \( g(x) = 4x^2 – 5 \) a ʻo ka hana o waho ʻo \( f(g) = \ln(g) \).
Ua ʻike mākou:
1. \( g'(x) = 8x \)
2. \( f'(g) = \frac{1}{g} \)
Ma ke kānāwai kaulahao, loaʻa iā mākou:
\[ y' = f'(g(x)) \cdot g'(x) = \frac{1}{4x^2 – 5} \cdot 8x \]
No laila, ʻo ke kumu o \( y = \ln(4x^2 – 5) \) penei:
\[ y' = \frac{8x}{4x^2 – 5} \]
Laʻana 4:
Hāʻawi ʻia ka hana \( y = (3x^2 + 2x + 1)^4 \), e huli i kāna derivative.
Kūkākūkā:
ʻO ka hana o loko ʻo \( g(x) = 3x^2 + 2x + 1 \) a ʻo ka hana o waho ʻo \( f(g) = g^4 \).
Ua ʻike mākou:
1. \( g'(x) = 6x + 2 \)
2. \( f'(g) = 4g^3 \)
Ma ke kānāwai kaulahao, loaʻa iā mākou:
ʻO ka huaʻōlelo "y' = f'(g(x)) g'(x) = 4(3x^2 + 2x + 1)^3 (6x + 2)
No laila, ʻo ke kumu o \( y = (3x^2 + 2x + 1)^4 \) penei:
\[ y' = 4(3x^2 + 2x + 1)^3 (6x + 2) \]
4. Nā Hihia Kūikawā a me ka Hoʻomohala ʻana o nā Rula Kaulahao
I kekahi manawa, ʻaʻole i pau ke kānāwai kaulahao ma ka haku mele ʻana o ʻelua mau hana wale nō. Aia kekahi mau manawa i ka wā he haku mele ʻana o nā hana ʻelua a ʻoi aku, no ka laʻana: \( h(x) = f(g(k(x))) \).
No ke ʻano o ʻekolu mau hana, hiki ke hoʻopili ʻia ke kānāwai kaulahao ma nā papa:
\[ h'(x) = f'(g(k(x))) \cdot g'(k(x)) \cdot k'(x) \]
Hiki iā mākou ke ʻike i kēlā me kēia papa, ke helu nei mākou i nā hopena o nā papa o waho ma mua o ka neʻe ʻana i nā hopena o nā papa o loko.
Laʻana 5:
Hāʻawi ʻia \( y = \sqrt{\ln(2x^2 + 1)} \), e huli i kāna derivative.
Kūkākūkā:
ʻO ka hana o loko loa ʻo \( k = 2x^2 + 1 \), waena: \( g = \ln(k) \) a me waho: \( f = \sqrt{g} \).
Ua ʻike mākou:
1. \( k'(x) = 4x \)
2. \( g'(k) = \frac{1}{k} \)
3. \( f'(g) = \frac{1}{2\sqrt{g}} \)
E hoʻopili kākou i ke kānāwai kaulahao ma nā papa:
\[ y' = f'(g(k(x))) \cdot g'(k(x)) \cdot k'(x) = \frac{1}{2\sqrt{\ln(2x^2 + 1)}} \cdot \frac{1}{2x^2 + 1} \cdot 4x \]
No laila, ʻo ke kumu o \( y = \sqrt{\ln(2x^2 + 1)} \) penei:
\[ y' = \frac{4x}{2(2x^2 + 1)\sqrt{\ln(2x^2 + 1)}} \]
5. Manaʻo
He kuleana koʻikoʻi ko ke kānāwai kaulahao i ka calculus differential, ʻoiai i ka wā e pili ana i ka haku mele ʻana o nā hana. ʻO ka hoʻomaopopo ʻana a me ka hoʻokele ʻana i ke kānāwai kaulahao e hāʻawi i kahi kahua paʻa no ka hoʻoponopono ʻana i nā pilikia paʻakikī i ka calculus. Ua kūkākūkā kēia ʻatikala i kekahi mau laʻana koʻikoʻi e hāʻawi i kahi ʻike paʻa o ka hoʻopili ʻana o ke kānāwai kaulahao i nā derivatives. Manaʻolana mākou he kōkua kēia kūkākūkā no nā haumāna a hiki ke hoʻopili ʻia i nā ʻano kūlana makemakika paʻakikī like ʻole.
