Bases zauv tsom xam. Yuav ua li cas mus nrhiav tau cov derivative?

Derivative ntawm ib tug muaj nuj nqi f (x) nyob rau hauv ib lub point x0 muaj nuj nqi hu ua txoj kev loj hlob piv txwv rau cov increment ntawm lub cav, muab hais tias x yuav 0, thiab tus ciam tshwm sim. Derivative feem ntau xaiv stroke, tej zaum ntawm taw tes los yog ntawm ib tug txawv. Feem ntau, cov derivative ntawm tus ntoo khaub lig-ciam teb yog yuam kev tau, txij li thaum xws li ib tug sawv cev yog tsis tshua muaj siv.

Muaj nuj nqi, uas muaj cov derivative hauv ib lub point x0, hu ua differentiable nyob rau hauv xws li ib tug taw tes. Xav hais tias, D1 - ib tug plurality ntawm cov ntsiab lus uas cov kev ua f yog txawv. Muab rau txhua ib ntawm cov zauv x, teej tug D f '(x), peb muab tau cov nuj nqi los cheeb tsam D1. Qhov no muaj nuj nqi yog derivative ntawm y = f (x). Yog xws li: f '(x).

Tsis tas li ntawd, cov derivative kheev siv nyob rau hauv physics thiab engineering. Xav txog ib tug piv txwv yooj yim. Cov khoom point tsiv rau ib tug tswj axis, thaum nug dab tsi raws li txoj cai ntawm cov lus tsa suab, uas yog, x-coordinate ntawm no taw tes yog lub npe hu x (t) muaj nuj nqi. Thaum lub sij hawm lub sij hawm luv los ntawm t0 rau t0 + t sib npaug rau hauv lub xaav ntawm tus taw tes x (t0 + t) -hluav (t0) = x, thiab nws nruab nrab ceev v (t) sib npaug zos rau x / t.

Tej zaum cov xwm ntawm cov lus tsa suab hais li ntawd hais tias qhov nruab nrab ceev tsis hloov thaum me me sij hawm Team sib, lub ntsiab lus uas zog nrog ib tug ntau dua degree ntawm cov neeg yog xav tau niaj hnub. Xwb, cov nqi ntawm qhov nruab nrab ceev yog tias t0 li nram no rau ib co kiag li leej nqi, thiab yog raug xa mus rau raws li lub instantaneous ceev v (t0) uas taw tes rau ib tug lub sij hawm ntawd lub sij hawm t0. Nws yog ntseeg hais tias cov instantaneous ceev v (t) yog lub npe hu rau tej differentiated muaj nuj nqi x (t), nyob li cas v (t) yog sib npaug zos rau x '(t). Cias muab, tus ceev - nws yog ib tug derivative ntawm lub coordinates ntawm lub sij hawm.

Instantaneous tshaj tawm muaj ob zoo thiab tsis zoo qhov tseem ceeb, thiab tus nqi yog 0. Yog hais tias nws yog thaum lub sij hawm luv (t1; t2) yog zoo, ces tus taw tes tsiv nyob rau hauv tib txoj kev coj, i.e., x (t) tswj tsub kom nrog lub sij hawm, thiab yog hais tias v (t) yog tsis zoo, ces tus tswj x (t) tsawg.

Nyob rau hauv ntau txoj mob, tus taw tes tsiv nyob rau hauv lub dav hlau los yog nyob rau hauv qhov chaw. Ces tus tshaj tawm los ntawm - ib tug vector kom muaj nuj nqis, thiab txiav txim rau txhua lub coordinates ntawm ib tug vector v (t).

Ib yam li ntawd, ib tug yuav piv tau rau acceleration ntawm tus taw tes. Ceev yog ib tug muaj nuj nqi ntawm lub sij hawm, piv txwv li, v = v (t). Ib tug derivative ntawm xws li ib tug muaj nuj nqi - tsab ntawv tsa suab acceleration: a = v '(t). Hais tias yog, nws hloov tawm hais tias lub sij hawm derivative ntawm ceev yog acceleration.

Piv txwv y = f (x) - tej differentiated muaj nuj nqi. Tom qab ntawd peb yuav xav txog cov lus tsa suab ntawm ib tug taw tes nyob rau hauv lub coordinate axis, uas yuav siv sij hawm qhov chaw rau cov kev cai lij choj x = f (t). Neeg kho tshuab txij nkawm ntawm lub derivative muab lub sij hawm los mus muab ib tug ntshiab txhais lus ntawm cov theorems ntawm lub differential calculus.

Yuav ua li cas mus nrhiav tau cov derivative? Nrhiav cov derivative ntawm ib tug muaj nuj nqi no yog hu ua nws cov ntau yam.

Muab koj cov piv txwv ntawm yuav ua li cas nrhiav tau cov derivative ntawm cov nuj nqi:

Lub derivative ntawm ib qhov muaj nuj nqi sib npaug zos rau xoom; derivative ntawm cov nuj nqi y = x yog sib npaug zos rau kev sib sau.

Thiab yuav ua li cas nrhiav tau cov derivative ntawm lub feem? Ua li no, xav txog cov nram cov ntaub ntawv uas:

Rau tej x0 <> 0 peb muaj

y / x = -1 / x0 * (x + x)

Muaj ib co kev cai, yuav ua li cas nrhiav tau cov derivative. Namely:

Yog hais tias lub zog A thiab B yog differentiated point x0, ces lawv sum yog txawv ntawm ib tug taw tes: (A + B) '= A' + B '. Cias muab, lub derivative ntawm ib tug sum sib npaug zos rau cov sum ntawm cov derivatives. Yog hais tias cov nuj nqi yog txawv ntawm ib co taw tes, ces nws yuav tsum increment rau xoom thaum tom qab lub sib cav mus pes tsawg nce.

Yog hais tias lub zog A thiab B yog differentiated point x0, ces lawv cov khoom yog txawv ntawm: (A * B) '= A'B + AB'. (Qhov tseem ceeb functions thiab lawv derivatives yog xam ntawm tus taw tes x0). Yog hais tias cov nuj nqi A (x) yog txawv nyob rau hauv point x0, thiab C - tsis tu ncua, ces CA muaj nuj nqi yog txawv nyob rau hauv qhov point thiab (CA) '= CA'. Uas yog, yog ib qhov zoo tshaj yuav coj sab nraum lub kos npe rau ntawm lub derivative.

Yog hais tias lub zog A thiab B yog differentiated point x0, thiab cov nuj nqi B yog tsis sib npaug zos rau zero, ces lawv piv kuj txawv ntawm: (A / B) '= (A'B-AB') / B * B.