Derivatives tus xov tooj: xam xyuas txoj kev thiab piv txwv

Tej zaum lub tswvyim ntawm derivative yog paub rau tag nrho cov ntawm peb txij thaum high school. Feem ntau cov tub ntxhais kawm muaj teeb meem to taub qhov no yog undoubtedly ib qho tseem ceeb heev tshaj plaws. Nws yog kev koom siv nyob rau hauv ntau qhov chaw ntawm tib neeg lub neej, thiab ntau yam xws li tshuab tau raws li precisely rau zauv muab los ntawm qhov derivative. Tab sis ua ntej tswv yim rau kev tsom xam ntawm dab tsi yog ib tug derivative ntawm tus xov tooj raws li lawv xam thiab qhov uas lawv yuav tuaj nyob rau hauv muab rau, delve ib tug me ntsis rau hauv keeb kwm.

zaj dabneeg

Lub tswvyim ntawm derivative, uas yog lub hauv paus ntawm zauv tsom xam, yog qhib (txawm zoo dua hais tias "yees" vim hais tias nws yog, raws li xws li, tsis muaj nyob rau hauv cov xwm) Isaakom Nyutonom, uas peb txhua tus paub los ntawm lub foundations ntawm cov kev cai lij choj ntawm lub ntiajteb txawj nqus. Nws yog nws leej twg ua ntej siv cov tswvyim no nyob rau hauv physics rau lub losis tswvyim dabtsi xwm ntawm cov kev ceev thiab acceleration ntawm lub cev. Thiab muaj ntau zaum tseem qhuas Newton rau no zoo kawg li invention, vim hais tias nyob rau hauv qhov tseeb nws invented lub hauv paus ntawm differential thiab integral calculus, lub tseeb lub hauv paus ntawm tag nrho cov teb cov kev kawm txog zauv hu ua "lej tsom xam". Seb thaum lub sij hawm lub Nobel nqi zog, Newton yuav yuav tau txais nws ib ob peb lub sij hawm.

Tsis yog tsis muaj lwm yam zoo kawg siab. Nyob rau hauv tas li ntawd mus Newton rau txoj kev loj hlob ntawm derivative thiab ib ua hauj lwm xws eminent geniuses ntawm kev kawm txog zauv raws li Leonhard Euler, Lagrange thiab Louis Gotfrid Leybnits. Nws yog ua tsaug rau lawv peb muaj lub hom phiaj ntawm differential calculus nyob rau hauv daim ntawv nyob rau hauv uas nws tshwm sim rau hnub no. Incidentally, qhov no yog Leibniz sab geometric ntsiab lus ntawm cov derivative, uas yog tsis muaj dab tsi ntau tshaj li tus nqes hav ntawm lub tangent rau lub teeb ntawm cov nuj nqi.

Yuav ua li cas yog ib tug derivative ntawm tus xov tooj? Ntsis rov dab tsi coj qhov chaw nyob rau hauv lub tsev kawm ntawv.

Yuav ua li cas yog ib tug derivative?

Txhais cov tswvyim no rau hauv ob peb txoj kev sib txawv. Qhov yooj yim piav qhia: derivatives - nws yog tus nqi ntawm kev hloov muaj nuj nqi. Sawv cev rau lub teeb ntawm tej kev ua y ntawm x. Yog hais tias nws yog tsis ncaj, nws muaj ib co curves nyob rau hauv lub teeb, lub sij hawm ntawm kev nce thiab txo. Yog hais tias koj siv sij hawm twg infinitesimal luv ntawm lub sij hawm, nws yuav tsum yog ib tug ncaj kab ya. Yog li ntawd, qhov ratio ntawm qhov luaj li cas ntawm ib infinitesimal ya ntawm lub y rau qhov loj ntawm lub x tswj, thiab yuav tsum yog ib tug derivative ntawm cov nuj nqi ntawm ib tug muab point. Yog hais tias peb xav txog qhov muaj nuj nqi raws li ib tug tag nrho, es hauv ib lub point, peb nrhiav tau ib tug muaj nuj nqi ntawm lub derivative, piv txwv li ib tug tej yam nyob tos rau lub X y.

Nyob rau hauv tas li ntawd, sib nrug los ntawm lub cev lub ntsiab lus ntawm lub derivative raws li ib tug muaj nuj nqi ntawm tus nqi ntawm kev hloov, muaj kuj yog ib tug geometric kev txiav txim zoo. Nyob rau nws, peb tam sim no tham txog.

Cov duab ntsiab lus

Derivatives xov tooj lawv tus kheej yog ib tug tej yam muaj pes tsawg tus uas tsis yog ib tug zoo siab tsis muaj tej yam ntsiab lus. Nws hloov tawm hais tias lub derivative yog tsis tsuas qhia tau hais tias txoj kev loj hlob npaum li cas los yog yuav txo cov nuj nqi, thiab tus nqes hav ntawm lub tangent rau lub teeb ntawm cov nuj nqi ntawm tus taw tes. Tsis yog tag nrho ntshiab txhais. Cia peb kawm nws nyob rau hauv kom meej. Piv txwv peb muaj ib lub teeb ntawm ib tug muaj nuj nqi (coj paj nkhaus). Nws muaj ib tug infinite tus naj npawb ntawm cov ntsiab lus, tab sis muaj tej qhov chaw uas tsuas muaj ib zaug xwb point muaj ib tug tshaj plaws los yog yam tsawg kawg nkaus. Los ntawm tej yam taw tes, koj muaj peev xwm kos ib tug ncaj kab, uas yuav perpendicular mus rau lub teeb ntawm cov nuj nqi ntawm tus taw tes. Qhov no kab yuav raug hu ua ib tug tangent. Piv txwv peb tuav nws mus txog rau lub tshuam nrog lub axis twm. Yog li ntawd tau ntawm lub tangent thiab lub axis twm thiab lub yuav tsum tau txiav txim los ntawm lub derivative. Dua, lub tangent ntawm no lub yuav ua ncaj rau nws.

Wb tham me ntsis txog cov neeg mob thiab derivatives Cia peb kawm cov zauv.

Tshwj xeeb mob

Raws li peb twb hais, derivatives ntawm tus xov tooj - ib tug derivative muaj nqis nyob rau ib qho kev taw tes. Ntawm no, piv txwv li, siv rau cov kev ua y = x ^2. Lub derivative ntawm x - xov tooj, tab sis nyob rau hauv kev - ib tug muaj nuj nqi sib npaug zos rau 2 * x. Yog hais tias peb yuav tsum tau los mus laij lub derivative, piv txwv li, nyob rau lub point x ₀ = 1, peb tau txais y '(1) = 2 * 1 = 2. Nws yog heev yooj yim. Ib tug nthuav cov ntaub ntawv yog tus derivative ntawm lub complex tooj. Yuav mus rau hauv ib tseeb piav li cas ib tug complex tooj, peb yuav tsis tau. Suffice nws hais tias qhov no muaj pes tsawg tus uas muaj lub thiaj li hu ua xav txog tej yam unit - lub xov tooj uas nws square sib npaug -1. Lub xam ntawm no derivative no tsuas tau nyob rau hauv cov nram no tej yam kev mob:

1) Yuav tsum muaj thawj thiaj li ib nrab derivatives ntawm qhov tiag tiag thiab xav txog tej yam qhov chaw ntawm y thiab X.

2) cov tej yam kev mob ntawm lub Cauchy-Riemann txuam nrog koob pheej ntawm lawv ib nrab piav nyob rau hauv thawj pawg lus.

Lwm nthuav cov ntaub ntawv, tab sis tsis yog li nyuab li yav dhau los ib tug, yog ib tug derivative ntawm ib tug tsis zoo pes tsawg. Nyob rau hauv qhov tseeb, tej cov zauv negative yuav tuaj raws li ib tug zoo, multiplied by -1. Zoo, tus derivative thiab lub qhov muaj nuj nqi sib npaug zos rau ib qhov khoo los ntawm cov derivative ntawm cov nuj nqi.

Nws yuav nthuav kom paub txog lub luag hauj lwm ntawm derivatives nyob rau hauv lawv lub neej txhua hnub, thiab qhov no yog tam sim no thiab sib tham txog nws.

daim ntawv thov

Tej zaum peb txhua tus tsawg kawg ib zaug nyob rau hauv ib tug lub neej txais kuv tus kheej xav tias kawm txog zauv yog tsis zoo li yuav pab tau rau nws. Thiab xws li ib tug tham tshaj plaws li lub derivative tej zaum twb tsis muaj kev siv. Nyob rau hauv qhov tseeb, lub lej - yuav tsum muaj kev kawm, thiab tag nrho nws cov txiv hmab txiv ntoo muaj tseem ceeb physics, Science News for KIDS, astronomy thiab txawm lub kev khwv nyiaj txiag. Derivative cim pib ntawm zauv tsom xam, uas tau muab rau peb lub sijhawm kos kev xaus ntawm lub graphs zog, thiab peb tau kawm los txhais lus rau cov kev cai ntawm qhov thiab tig lawv mus rau lawv kom zoo dua vim hais tias ntawm nws.

xaus

Ntawm cov hoob kawm, tsis yog txhua tus yuav ua tau pab tau rau lub derivative nyob rau hauv lub neej tiag tiag. Tab sis kev ua lej npaj logic uas yuav muaj tseeb xav tau. Tsis rau ib yam dab tsi vim hais tias kev kawm txog zauv yog hu ua tus huab tais ntawm sciences: nws muaj ib tug to taub yooj yim ntawm lwm teb ntawm kev txawj ntse.