Kev Sau Cov Derivative ntawm Ib Lub Function
Pendahuluuan
Hauv kev lej, tshwj xeeb tshaj yog calculus, cov derivative yog ib lub tswv yim tseem ceeb uas ua lub luag haujlwm tseem ceeb hauv ntau yam kev siv. Cov derivatives tsis yog siv rau hauv kev lej theoretical xwb tab sis kuj siv rau hauv kev tshawb fawb, engineering, economics, thiab ntau lwm yam kev kawm. Tsab xov xwm no yuav tham txog cov derivative ntawm ib qho function kom ntxaws, npog nws cov hauv paus, cov cai tseem ceeb, thiab cov piv txwv ntawm kev siv.
Cov Ntsiab Lus ntawm Derivatives
Kev Txhais ntawm Cov Khoom Siv Derivatives
Tus derivative ntawm ib qho function piav qhia txog qhov kev hloov pauv ntawm qhov function nrog rau nws cov variable ywj pheej. Intuitively, tus derivative tuaj yeem txhais tau tias yog qhov nqes hav ntawm txoj kab tangent uas kov daim duab ntawm qhov function ntawm ib qho chaw.
Yog tias \( y = f(x) \), ces thawj qhov derivative ntawm \( f \) nrog rau \( x \) yog qhia los ntawm \( f'(x) \) lossis \( \frac{dy}{dx} \). Lub ntsiab lus raug cai ntawm qhov derivative yog muab los ntawm cov kev txwv hauv qab no:
\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) – f(x)}{h} \]
Cov Cim Qhia Txog Kev Siv Derivative
Muaj ntau cov cim qhia uas feem ntau siv rau hauv kev sau cov derivatives:
1. Leibniz cov cim qhia: \( \frac{dy}{dx} \)
2. Lagrange notation: \( f'(x) \)
3. Newton cov cim qhia: \( y' \)
4. Euler cim: \( Df(x) \)
Txhua daim ntawv sau muaj cov kev siv tshwj xeeb thiab cov ntsiab lus uas lawv siv ntau dua.
Cov Cai Tseem Ceeb hauv Kev Sib Txawv
Cov Cai Ntxiv thiab Rho Tawm
Yog tias \( f(x) \) thiab \( g(x) \) yog ob qho kev ua haujlwm sib txawv, ces:
\[ \frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x) \]
Cov Cai ntawm Kev Sib Npaug
Rau ob lub luag haujlwm \( u(x) \) thiab \( v(x) \):
\[ \frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) \]
Cov Cai ntawm Kev Faib
Yog tias \( u(x) \) thiab \( v(x) \) yog ob qho kev ua haujlwm, thiab \( v(x) \neq 0 \):
\[ \frac{d}{dx} \left[ \frac{u(x)}{v(x)} \right] = \frac{u'(x) \cdot v(x) – u(x) \cdot v'(x)}{[v(x)]^2} \]
Txoj Cai Saw
Rau qhov sib xyaw ua ke ntawm ob lub luag haujlwm \( f(u) \) thiab \( u(g) \):
\[ \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) \]
Piv txwv ntawm Daim Ntawv Thov
Cov Kev Sib Piv ntawm Cov Kev Ua Haujlwm Polynomial
Xav tias \( f(x) = 3x^3 – 5x^2 + 2x – 1 \). Txhawm rau nrhiav qhov derivative ntawm qhov kev ua haujlwm no, peb siv cov cai yooj yim ntawm kev sib txawv.
\[ f'(x) = \frac{d}{dx} (3x^3) – \frac{d}{dx} (5x^2) + \frac{d}{dx} (2x) – \frac{d}{dx} (1) \]
\[ f'(x) = 9x^2 – 10x + 2 \]
Cov Kev Sib Piv ntawm Exponential thiab Logarithmic Functions
Yog tias \( f(x) = e^x \), ces qhov derivative ntawm exponential function yog:
\[ f'(x) = e^x \]
Rau qhov kev ua haujlwm logarithm ntuj \( f(x) = \ln(x) \):
\[ f'(x) = \frac{1}{x} \]
Cov Kev Sib Piv ntawm Trigonometric Functions
Rau cov haujlwm trigonometric yooj yim:
- Yog tias f(x) = sin(x) \, ces f'(x) = cos(x) \)
- Yog tias f(x) = cos(x) \), ces f'(x) = -sin(x) \)
- Yog tias \( f(x) = \tan(x) \), ces \( f'(x) = \sec^2(x) \)
Kev Txheeb Xyuas ntawm Kev Ua Haujlwm Sib Xyaws
Xav tias \( f(x) = \sin(2x) \). Peb tuaj yeem siv txoj cai saw hlau:
\[ f'(x) = \cos(2x) \cdot \frac{d}{dx}(2x) = \cos(2x) \cdot 2 = 2\cos(2x) \]
Cov Khoom Siv Derivatives Siab Tshaj Plaws
Cov Khoom Siv Thib Ob thiab Tom Qab
Tus derivative thib ob yog tus derivative ntawm thawj derivative function. Yog tias \( y = f(x) \) ces tus derivative thib ob yog cim los ntawm \( f”(x) \) lossis \( \frac{d^2y}{dx^2} \). Thiab yog li ntawd rau tus derivative thib peb \( f”'(x) \) lossis \( \frac{d^3y}{dx^3} \).
Xav tias \( f(x) = x^4 \):
\[ f'(x) = 4x^3 \]
\[ f”(x) = \frac{d}{dx}(4x^3) = 12x^2 \]
\[ f”'(x) = \frac{d}{dx}(12x^2) = 24x \]
\[ f””(x) = \frac{d}{dx}(24x) = 24 \]
Cov Kev Siv ntawm Derivatives hauv Physics
Hauv kev kawm txog physics, cov derivatives feem ntau siv los txiav txim siab qhov ceev thiab kev nrawm. Xav tias \( s(t) \) yog ib qho kev ua haujlwm ntawm txoj haujlwm piv rau lub sijhawm \(t \). Qhov ceev \( v(t) \) yog thawj qhov derivative ntawm txoj haujlwm:
\[ v(t) = s'(t) \]
Kev nrawm \( a(t) \) yog thawj qhov derivative ntawm qhov ceev lossis qhov thib ob derivative ntawm qhov chaw:
\[ a(t) = v'(t) = s”(t) \]
Xaus
Tus derivative ntawm ib qho function yog ib lub tswv yim tseem ceeb hauv calculus uas muaj kev siv dav dav thoob plaws ntau qhov chaw. Kev nkag siab txog tus derivative ua tus nqes hav ntawm ib txoj kab tangent muab kev nkag siab tseem ceeb rau cov khoom thiab tus cwj pwm ntawm ib qho function. Kev nkag siab thiab kev muaj peev xwm siv cov cai ntawm kev sib txawv xws li txoj cai saw, txoj cai khoom, thiab txoj cai faib yog qhov tseem ceeb rau txhua tus neeg kawm calculus. Los ntawm cov piv txwv yooj yim thiab cov ntawv thov hauv physics, tsab xov xwm no vam tias yuav muab kev nkag siab dav dav txog kev sau tus derivative ntawm ib qho function.
