Differentiation vs Derivative
Nyob rau hauv cov lej sib txawv, qhov sib txawv thiab qhov sib txawv yog qhov sib thooj, tab sis txawv heev, thiab siv los sawv cev rau ob lub ntsiab lus tseem ceeb ntawm kev ua lej ntsig txog kev ua haujlwm.
Dab tsi yog derivative?
Derivative ntawm kev ua haujlwm ntsuas tus nqi ntawm qhov kev ua haujlwm tus nqi hloov pauv raws li nws cov tswv yim hloov pauv. Hauv kev ua haujlwm ntau yam sib txawv, qhov kev hloov pauv ntawm tus nqi muaj nuj nqi nyob ntawm qhov kev taw qhia ntawm qhov kev hloov pauv ntawm qhov tseem ceeb ntawm cov kev hloov pauv ywj pheej. Yog li ntawd, nyob rau hauv cov xwm txheej zoo li no, ib qho kev taw qhia tshwj xeeb raug xaiv thiab kev ua haujlwm sib txawv hauv qhov kev taw qhia tshwj xeeb. Qhov derivative no hu ua directional derivative. Cov derivatives ib nrab yog ib yam tshwj xeeb ntawm cov kev taw qhia derivatives.
Derivative ntawm vector-valued muaj nuj nqi f tuaj yeem txhais tau tias yog qhov txwv [latex]\\ frac{df}{d\\boldsymbol{u}}=\\lim_{h \to 0}\\ frac {f(\boldsymbol{x}+h \\boldsymbol{u})-f(\boldsymbol{x})}{h}[/latex] nyob qhov twg nws muaj finitely. Raws li tau hais ua ntej, qhov no muab peb tus nqi nce ntawm kev ua haujlwm f raws kev coj ntawm vector u. Nyob rau hauv cov ntaub ntawv ntawm ib tug muaj nuj nqis muaj nuj nqi, qhov no txo mus rau lub zoo-paub txhais ntawm cov derivative, [latex]\\ frac{df}{dx}=\\lim_{h \\ to 0}\\ frac{f (x+h)-f(x)}{h}[/latex]
Piv txwv li, [latex]f(x)=x^{3}+4x+5[/latex] yog txhua qhov sib txawv, thiab cov derivative yog sib npaug rau qhov txwv, [latex]\\lim_{h \\to 0}\\frac{(x+h)^{3}+4(x+h)+5-(x^{3}+4x+5)}{h}[/latex], uas yog sib npaug rau [latex]3x^{2}+4[/latex]. Cov derivatives ntawm kev ua haujlwm xws li [latex]e^{x}, \\sin x, \\cos x[/latex] muaj nyob txhua qhov chaw. Lawv feem sib npaug rau cov haujlwm [latex]e^{x}, \\cos x, – \\sin x[/latex].
Qhov no yog lub npe hu ua thawj derivative. Feem ntau cov thawj derivative ntawm kev ua f yog denoted los ntawm f (1) Tam sim no siv cov cim no, nws muaj peev xwm los txhais cov kev txiav txim siab dua. [latex]\\frac{d^{2}f}{dx^{2}}=\\lim_{h \\to 0}\\frac{f^{(1)}(x+h)-f ^{(1)}(x)}{h}[/latex] is the second order directional derivative, and denoting the n th derivative by f (n) rau txhua n, [latex]\\frac{d^{n}f}{dx^{n}}=\\lim_{h \\to 0}\\frac{f^{(n -1)}(x+h)-f^{(n-1)}(x)}{h}[/latex], txhais cov n th derivative.
Dab tsi yog qhov txawv?
Differentiation yog tus txheej txheem ntawm kev nrhiav cov derivative ntawm ib qho kev sib txawv. D-tus neeg ua haujlwm qhia los ntawm D sawv cev rau kev sib txawv hauv qee qhov ntsiab lus. Yog tias x yog qhov hloov pauv ywj pheej, ces D ≡ d / dx. Tus neeg ua haujlwm D-tus neeg teb xov tooj yog tus neeg teb xov tooj, piv txwv li rau ob qho kev ua haujlwm sib txawv f thiab g thiab tas li c, raws li cov khoom tuav.
I. D (f + g)=D (f) + D(g)
II. D (cf)=cD (f)
Siv tus neeg ua haujlwm D, lwm cov cai cuam tshuam nrog kev sib txawv tuaj yeem qhia tau raws li hauv qab no. D (f g)=D (f) g + f D (g), D (f/ g)=[D (f) g – f D (g)]/ g 2 and D (f o g)=(D (f) o g) D(g).
Piv txwv li, thaum F(x)=x 2sin x sib txawv ntawm kev hwm x siv cov cai muab, cov lus teb yuav yog 2 x sin x + x2cos x.
Dab tsi yog qhov sib txawv ntawm qhov sib txawv thiab qhov sib txawv?• Derivative hais txog tus nqi ntawm kev hloov pauv ntawm kev ua haujlwm • Kev sib txawv yog tus txheej txheem ntawm kev nrhiav cov txiaj ntsig ntawm kev ua haujlwm. |
