Cov Lus Nug Piv Txwv Sib Tham Txog Kev Sau Cov Kev Ua Haujlwm Derivatives
Tus derivative ntawm ib qho function yog ib lub tswv yim tseem ceeb hauv calculus, feem ntau siv rau ntau qhov chaw ntawm kev tshawb fawb, xws li physics, economics, biology, thiab engineering. Tus derivative ntawm ib qho function ntsuas seb nws tus nqi hloov pauv sai npaum li cas piv rau kev hloov pauv hauv nws cov variables ywj pheej. Hauv tsab xov xwm no, peb yuav tham txog ntau qhov piv txwv ntawm cov teeb meem uas cuam tshuam nrog kev sau tus derivative ntawm ib qho function, ua tiav nrog cov lus piav qhia.
Piv txwv lus nug 1: Derivative ntawm Simple Functions
Lus Nug: Nrhiav thawj qhov derivative ntawm lub function \( f(x) = 3x^2 + 5x + 7 \).
Kev Sib Tham:
Txhawm rau txiav txim siab thawj qhov derivative ntawm ib qho kev ua haujlwm \( f(x) \), peb siv cov cai yooj yim ntawm kev sib txawv, uas yog:
\[
\frac{d}{dx}(ax^n) = anx^{n-1}
\]
Yog li, peb tuaj yeem xam qhov derivative ntawm txhua lub sijhawm hauv qhov kev ua haujlwm raws li hauv qab no:
\[
f'(x) = \frac{d}{dx}(3x^2) + \frac{d}{dx}(5x) + \frac{d}{dx}(7)
\]
\[
f'(x) = 3 x^{2-1} + 5 x^{1-1} + 0
\]
\[
f'(x) = 6x + 5
\]
Yog li, thawj qhov derivative ntawm qhov kev ua haujlwm \( f(x) = 3x^2 + 5x + 7 \) yog \( f'(x) = 6x + 5 \).
Piv txwv lus nug 2: Cov derivatives ntawm Trigonometric Functions
Lus Nug: Nrhiav thawj qhov derivative ntawm qhov kev ua haujlwm \( g(x) = \sin(x) + \cos(x) \).
Kev Sib Tham:
Peb siv cov cai yooj yim rau cov haujlwm trigonometric:
\[
\frac{d}{dx}(\sin(x)) = \cos(x)
\]
\[
\frac{d}{dx}(\cos(x)) = -\sin(x)
\]
Yog li ntawd:
\[
g'(x) = \frac{d}{dx}(\sin(x)) + \frac{d}{dx}(\cos(x))
\]
\[
g'(x) = \cos(x) - \sin(x)
\]
Yog li, thawj qhov derivative ntawm qhov kev ua haujlwm \(g(x) = \sin(x) + \cos(x) \) yog \(g'(x) = \cos(x) - \sin(x) \).
Piv txwv lus nug 3: Derivative ntawm Multiplication Function
Lus Nug: Nrhiav thawj qhov derivative ntawm lub function \( h(x) = x^2 \sin(x) \).
Kev Sib Tham:
Rau cov functions uas yog cov khoom ntawm ob lub functions, peb siv txoj cai multiplication:
\[
\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
\]
Xav tias \( u(x) = x^2 \) thiab \( v(x) = \sin(x) \). Ces:
\[
u'(x) = \frac{d}{dx}(x^2) = 2x
\]
\[
v'(x) = \frac{d}{dx}(\sin(x)) = \cos(x)
\]
Siv txoj cai sib npaug, peb tuaj yeem sau:
\[
h'(x) = [x^2]' \sin(x) + x^2 [\sin(x)]'
\]
\[
h'(x) = 2x \sin(x) + x^2 \cos(x)
\]
Yog li, thawj qhov derivative ntawm qhov kev ua haujlwm \(h(x) = x^2 \sin(x) \) yog \(h'(x) = 2x \sin(x) + x^2 \cos(x) \).
Piv txwv lus nug 4: Derivative ntawm Composition Function
Lus Nug: Nrhiav thawj qhov derivative ntawm qhov kev ua haujlwm \( k(x) = \sin(x^2) \).
Kev Sib Tham:
Rau cov functions uas yog cov sib xyaw ua ke ntawm ob lub functions, peb siv txoj cai saw hlau:
\[
\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)
\]
Cia \( f(u) = \sin(u) \) thiab \( u = x^2 \). Ces \( f'(u) = \cos(u) \) thiab \( g'(x) = \frac{d}{dx}(x^2) = 2x \).
Siv txoj cai saw hlau, peb tuaj yeem sau:
\[
k'(x) = \frac{d}{dx}[\sin(x^2)] = \cos(x^2)\cdot 2x
\]
Yog li, thawj qhov derivative ntawm qhov kev ua haujlwm \(k(x) = \sin(x^2) \) yog \(k'(x) = 2x \cos(x^2) \).
Piv txwv lus nug 5: Derivative ntawm Rational Functions
Teeb Meem: Nrhiav thawj qhov derivative ntawm qhov function \( m(x) = \frac{2x}{x^2 + 1} \).
Kev Sib Tham:
Rau cov functions uas yog quotient ntawm ob lub functions, peb siv txoj cai quotient:
\[
\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) – u(x)v'(x)}{[v(x)]^2}
\]
Xav tias \( u(x) = 2x \) thiab \( v(x) = x^2 + 1 \). Ces:
\[
u'(x) = 2
\]
\[
v'(x) = \frac{d}{dx}(x^2 + 1) = 2x
\]
Siv txoj cai quotient, peb tuaj yeem sau:
\[
m'(x) = \frac{[2x]'(x^2 + 1) – 2x[x^2 + 1]'}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2(x^2 + 1) – 2x^2}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2x^2 + 2 – 4x^2}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{-(2x^2 – 2)}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2 – 2x^2}{(x^2 + 1)^2}
\]
Yog li, thawj qhov derivative ntawm qhov kev ua haujlwm \( m(x) = \frac{2x}{x^2 + 1} \) yog \( m'(x) = \frac{2 – 2x^2}{(x^2 + 1)^2} \).
Xaus
Hauv tsab xov xwm no, peb tau tham txog ntau qhov piv txwv ntawm cov teeb meem uas cuam tshuam nrog cov derivatives ntawm cov functions, xws li cov functions yooj yim, trigonometric functions, multiplication, composition, thiab rational functions. Txhua qhov piv txwv qhia txog kev siv cov derivative rule, xws li cov fundamental rule, the chain rule, the multiplication rule, thiab the quotient rule. Kev nkag siab txog yuav ua li cas siv cov rule no yog qhov tseem ceeb rau kev daws cov teeb meem calculus nyuaj dua thoob plaws ntau yam kev qhuab qhia. Kev xyaum thiab kev cob qhia rov ua dua yuav pab txhawb koj txoj kev nkag siab thiab kev txawj ntse hauv kev sib txawv ntawm cov functions.
