Derivative vs Differential
Nyob rau hauv cov lej sib txawv, qhov sib txawv thiab qhov sib txawv ntawm ib qho kev ua haujlwm zoo sib xws tab sis muaj lub ntsiab lus sib txawv, thiab siv los sawv cev rau ob yam khoom siv lej tseem ceeb ntsig txog cov haujlwm sib txawv.
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?
Differential ntawm ib qho kev ua haujlwm sawv cev rau kev hloov pauv hauv kev ua haujlwm nrog rau cov kev hloov pauv ntawm tus kheej los yog hloov pauv. Hauv kev sau ib txwm muaj, rau ib qho kev ua haujlwm f ntawm ib qho kev sib txawv x, tag nrho qhov sib txawv ntawm kev txiav txim 1 df yog muab los ntawm, [latex]df=f^{1}(x)dx[/latex]. Qhov no txhais tau tias rau qhov kev hloov pauv tsis kawg hauv x (piv txwv li d x), yuav muaj f (1)(x)d x hloov hauv f.
Siv txwv ib qho tuaj yeem xaus nrog cov lus txhais raws li hauv qab no. Piv txwv tias ∆ x yog qhov kev hloov pauv hauv x ntawm qhov chaw tsis txaus ntseeg x thiab ∆ f yog qhov hloov pauv ntawm qhov ua haujlwm f. Nws tuaj yeem pom tau tias ∆ f=f (1)(x)∆ x + ϵ, qhov twg ϵ yog qhov yuam kev. Tam sim no, qhov txwv ∆ x→ 0∆ f / ∆ x =f (1)(x) (siv lub ntsiab txhais ua ntej ntawm derivative) thiab yog li, ∆ x→ 0 ϵ/ ∆ x=0. xaus tias, ∆ x → 0 ϵ=0. Tam sim no, denoting ∆ x → 0 ∆ f as d f thiab ∆ x → 0 ∆ x as d x lub ntsiab lus ntawm qhov sib txawv yog nruj heev.
Piv txwv li, qhov sib txawv ntawm kev ua haujlwm [latex]f(x)=x^{3}+4x+5[/latex] is [latex](3x^{2}+4)dx[/latex].
Rau cov haujlwm ntawm ob lossis ntau qhov sib txawv, tag nrho qhov sib txawv ntawm qhov muaj nuj nqi yog txhais raws li cov lej ntawm qhov sib txawv hauv cov lus qhia ntawm txhua qhov kev hloov pauv ywj pheej. Kev ua lej, nws tuaj yeem sau tau tias [latex]df=\\sum_{i=1}^{n} \frac{\partial f}{\partial x_{i}}dx_{i}[/latex].
Dab tsi yog qhov sib txawv ntawm qhov sib txawv thiab qhov txawv?
• Derivative yog hais txog tus nqi ntawm kev hloov pauv ntawm qhov kev ua haujlwm whereas qhov sib txawv yog hais txog qhov kev hloov pauv ntawm txoj haujlwm, thaum qhov kev hloov pauv ywj pheej raug hloov pauv.
• The derivative is muab los ntawm [latex]\\frac{df}{dx}=\\lim_{h \to 0}\\frac{f(x+h)-f(x)}{ h}[/latex], tab sis qhov sib txawv yog muab los ntawm [latex]df=f^{1}(x)dx[/latex].
