Cov Lus Nug Piv Txwv thiab Kev Sib Tham Txog Txoj Cai Chain hauv Derivatives
Txoj cai saw hlau yog ib qho ntawm cov tswv yim tseem ceeb tshaj plaws hauv kev suav lej sib txawv, siv los xam qhov derivative ntawm ib qho function uas muaj ob lossis ntau dua functions. Hauv tsab xov xwm no, peb yuav tham txog lub tswv yim tseem ceeb ntawm txoj cai saw hlau, yuav siv nws li cas, thiab piv txwv ntawm nws siv hauv cov teeb meem derivative uas feem ntau tshwm sim hauv tsev kawm ntawv theem siab thiab tsev kawm qib siab.
1. Kev Taw Qhia Txog Txoj Cai Saw
Ua ntej peb nkag mus rau hauv qhov teeb meem piv txwv, cia peb xub nkag siab txog txoj cai saw hlau yog dab tsi. Txoj cai saw hlau hais tias yog tias peb muaj ob lub luag haujlwm sib txawv \(f\) thiab \(g\), thiab peb xav nrhiav qhov derivative ntawm cov qauv ntawm cov haujlwm \(h = f(g(x))\), ces qhov derivative ntawm \(h\) yog:
\[ h'(x) = f'(g(x)) \cdot g'(x) \]
Hauv cov lus yooj yim, peb xam qhov derivative ntawm lub function sab nraud ntawm g(x), tom qab ntawd muab qhov tshwm sim los ntawm qhov derivative ntawm lub function sab hauv \(g(x)\).
2. Nkag Siab Txog Lub Luag Haujlwm ntawm Kev Sau Ntawv
Ua ntej peb nkag mus rau hauv cov teeb meem piv txwv, nws yog ib qho tseem ceeb kom nkag siab txog cov haujlwm sib xyaw. Ib qho haujlwm sib xyaw yog ib qho haujlwm tau los ntawm kev ntxig ib qho haujlwm rau hauv lwm qhov. Piv txwv li, yog tias peb muaj \( f(x) = \sin(x) \) thiab \( g(x) = x^2 \), ces qhov sib xyaw ntawm ob qho haujlwm yuav yog \( h(x) = f(g(x)) = \sin(x^2) \).
Hauv cov kev ua haujlwm sib sau ua ke, peb feem ntau xav txog \(g(x) \) ua "kev ua haujlwm sab hauv" thiab \(f(x) \) ua "kev ua haujlwm sab nraud". Hauv qhov piv txwv no, kev ua haujlwm sab hauv yog \(x^2 \) thiab kev ua haujlwm sab nraud yog sine.
3. Piv txwv cov lus nug thiab kev sib tham
Cia peb saib qee cov piv txwv ntawm cov teeb meem uas siv txoj cai saw hlau los daws lawv.
Piv txwv 1:
Muab qhov kev ua haujlwm \( y = \cos(3x^2) \), nrhiav thawj qhov derivative ntawm y piv rau x.
Kev Sib Tham:
Ua ntej, peb txheeb xyuas cov haujlwm sab hauv thiab sab nraud. Ntawm no, cov haujlwm sab hauv yog \(g(x) = 3x^2 \) thiab cov haujlwm sab nraud yog \(f(g) = \cos(g) \).
Peb paub:
1. \( g'(x) = 6x \)
2. \( f'(g) = -\sin(g) \)
Los ntawm txoj cai saw hlau, peb tau txais:
\[ y' = f'(g(x)) \cdot g'(x) = -\sin(3x^2) \cdot 6x \]
Yog li, qhov derivative ntawm \( y = \cos(3x^2) \) yog:
\[ y' = -6x \sin(3x^2) \]
Piv txwv 2:
Nrhiav thawj qhov derivative ntawm \( h(x) = e^{5x^3 + 2x} \).
Kev Sib Tham:
Nov qhov kev ua haujlwm sab hauv yog \(g(x) = 5x^3 + 2x \) thiab qhov kev ua haujlwm sab nraud yog \(f(g) = e^g \).
Peb paub:
1. \( g'(x) = 15x^2 + 2 \)
2. \( f'(g) = e^g \)
Los ntawm txoj cai saw hlau, peb tau txais:
\[ h'(x) = f'(g(x)) g'(x) = e^{5x^3 + 2x} \cdot (15x^2 + 2) \]
Yog li, qhov derivative ntawm \( h(x) = e^{5x^3 + 2x} \) yog:
\[ h'(x) = (15x^2 + 2)e^{5x^3 + 2x} \]
Piv txwv 3:
Nrhiav thawj qhov derivative ntawm \( y = \ln(4x^2 – 5) \).
Kev Sib Tham:
Lub luag haujlwm sab hauv yog \(g(x) = 4x^2 – 5 \) thiab lub luag haujlwm sab nraud yog \(f(g) = \ln(g) \).
Peb paub:
1. \( g'(x) = 8x \)
2. \( f'(g) = \frac{1}{g} \)
Los ntawm txoj cai saw hlau, peb tau txais:
\[ y' = f'(g(x)) g'(x) = \frac{1}{4x^2 – 5} \cdot 8x \]
Yog li, qhov derivative ntawm \( y = \ln(4x^2 – 5)\) yog:
\[ y' = \frac{8x}{4x^2 – 5} \]
Piv txwv 4:
Muab qhov kev ua haujlwm \( y = (3x^2 + 2x + 1)^4 \), nrhiav nws cov derivative.
Kev Sib Tham:
Lub luag haujlwm sab hauv yog \(g(x) = 3x^2 + 2x + 1 \) thiab lub luag haujlwm sab nraud yog \(f(g) = g^4 \).
Peb paub:
1. \( g'(x) = 6x + 2 \)
2. \( f'(g) = 4g^3 \)
Los ntawm txoj cai saw hlau, peb tau txais:
\[ y' = f'(g(x)) \cdot g'(x) = 4(3x^2 + 2x + 1)^3 \cdot (6x + 2) \]
Yog li, qhov derivative ntawm \( y = (3x^2 + 2x + 1)^4 \) yog:
\[ y' = 4(3x^2 + 2x + 1)^3 (6x + 2) \]
4. Cov Teeb Meem Tshwj Xeeb thiab Kev Tsim Cov Cai Saw
Qee zaum, txoj cai saw hlau tsis xaus rau ntawm kev sib xyaw ua ke ntawm ob lub luag haujlwm xwb. Muaj qee zaum thaum ib lub luag haujlwm yog kev sib xyaw ua ke ntawm ntau dua ob lub luag haujlwm, piv txwv li: \( h(x) = f(g(k(x))) \).
Rau cov xwm txheej ntawm peb txoj haujlwm, txoj cai saw hlau tuaj yeem siv rau hauv cov txheej:
\[ h'(x) = f'(g(k(x))) \cdot g'(k(x)) \cdot k'(x) \]
Peb tuaj yeem pom tias hauv txhua txheej, peb xam cov derivatives ntawm cov txheej sab nraud ua ntej txav mus rau cov derivatives ntawm cov txheej sab hauv.
Piv txwv 5:
Muab \( y = \sqrt{\ln(2x^2 + 1)} \), nrhiav nws cov derivative.
Kev Sib Tham:
Lub luag haujlwm sab hauv tshaj plaws yog \( k = 2x^2 + 1 \), nruab nrab: \( g = \ln(k) \) thiab sab nraud: \( f = \sqrt{g} \).
Peb paub:
1. \( k'(x) = 4x \)
2. \( g'(k) = \frac{1}{k} \)
3. \( f'(g) = \frac{1}{2\sqrt{g}} \)
Cia peb siv txoj cai saw hlau hauv cov txheej:
\[ y' = f'(g(k(x))) \cdot g'(k(x)) \cdot k'(x) = \frac{1}{2\sqrt{\ln(2x^2 + 1)}} \cdot \frac{1}{2x^2 + 1} \cdot 4x \]
Yog li ntawd, qhov derivative ntawm y = ...
\[ y' = \frac{4x}{2(2x^2 + 1)\sqrt{\ln(2x^2 + 1)}} \]
5. Txoj kev
Txoj cai saw hlau ua lub luag haujlwm tseem ceeb hauv kev suav lej sib txawv, tshwj xeeb tshaj yog thaum cuam tshuam nrog kev sib xyaw ua ke ntawm cov haujlwm. Kev nkag siab thiab kev paub txog txoj cai saw hlau muab lub hauv paus ruaj khov rau kev daws cov teeb meem nyuaj dua hauv kev suav lej. Tsab xov xwm no tau tham txog ntau qhov piv txwv tseem ceeb los muab kev nkag siab zoo txog kev siv txoj cai saw hlau rau cov derivatives. Peb vam tias qhov kev sib tham no yuav pab tau rau cov tub ntxhais kawm thiab tuaj yeem siv rau ntau yam xwm txheej lej nyuaj.
