Cov kev cai ntawm ntau yam, thov lej

Yuav pib, nws yog ib nqi nco ntsoov hais tias xws li differential thiab ib tug zauv ntsiab lus nws nqa.

Txawv muaj nuj nqi yog qhov khoom ntawm lub derivative muaj nuj nqi ntawm lub cav nyob rau hauv lub differential ntawm lub cav. Lej, cov tswvyim no yuav tau sau ntawv raws li ib qho kev qhia: dy = y '* dx.

Nyob rau hauv lem, los mus txiav txim lub derivative ntawm lub koob pheej ntawm lawv y '= Lim dx-0 (dy / dx), thiab los mus txiav txim cov kev txwv - tus qhia dy / dx = x' + α, qhov twg parameter α yog infinitesimal zauv kom muaj nuj nqis.

Yog li ntawd, ob tog ntawm cov kev qhia yuav tsum tau khoo los ntawm dx, uas thaum kawg muab dy = y '* dx + α * dx, qhov twg dx - yog ib infinitesimal hloov nyob rau hauv lub cav, (α * dx) - tus nqi ntawm uas yuav tsis saib xyuas, ces dy - increment functions, thiab (y * dx) - qhov tseem ceeb ntawm lub increment los yog differential.

Txawv muaj nuj nqi yog qhov khoom ntawm lub derivative muaj nuj nqi rau cov differential ntawm lub cav.

Tam sim no nws yog tsim nyog los xav txog cov kev cai ntawm ntau yam, uas yog feem ntau siv nyob rau hauv lej tsom xam.

Theorem. Derivative nqi sib npaug zos rau cov sum ntawm cov khoom uas tau los ntawm Cheebtsam: (a + c) = ib tug '+ c'.

Ib yam li ntawd, txoj cai no yuav ua kom nquag plias rau hauv lub derivative ntawm qhov sib txawv.
Lub qhov tsim nyog tau danogo cai ntawm ntau yam yog cov assertion tias cov derivative ntawm ib tug xov tooj ntawm cov nqe lus sib npaug zos rau cov sum ntawm cov khoom muab los ntawm cov lus.

Piv txwv li, yog tias koj xav mus nrhiav tus derivative ntawm cov kev qhia (a + c-k) ', ces cov no ib qho kev qhia ntawm ib tug' + c 'k'.

Theorem. Lub derivative khoom ntawm zauv zog differentiable ntawm ib tug taw tes sib npaug zos rau cov sum muaj cov khoom ntawm cov thawj tau mus rau lub thib ob derivative thiab cov khoom ntawm lub thib ob zoo tshaj mus rau tus thawj derivative.

Theorem yog lej sau raws li nram no: (a * c) '= ib tug * ib tug' + ib tug '* s. Qhov tsim nyog ntawm cov theorem yog ib tug xaus hais tias lub qhov zoo tshaj nyob rau hauv lub derivative ntawm cov khoom tej zaum yuav tsum tau noj sab nraum lub derivative muaj nuj nqi.

Nyob rau hauv daim ntawv ntawm ib tug algebraic qhia, txoj cai no yog sau raws li nram no: (a * c) = ib tug * ib tug ', qhov twg ib tug = const.

Piv txwv li, yog tias koj xav mus nrhiav tus derivative ntawm cov kev qhia (2a3) ', qhov tshwm sim yog cov lus teb: 2 * (A3) = 2 * 3 * 6 * a2 = a2.

Theorem. Derivative kev sib raug zoo zog sib npaug zos rau cov piv ntawm cov sib txawv ntawm cov derivative ntawm lub numerator khoo los ntawm lub denominator thiab cov numerator lub sij hawm lub derivative ntawm lub denominator thiab cov square ntawm lub denominator.

Theorem yog lej sau raws li nram no: (a / c) '= ( ib tug' * ib * a-c ') / ^2.

Nyob rau hauv xaus, nws yog tsim nyog los xav txog cov kev cai rau differentiating puas zog.

Theorem. Muab ib tug fuktsii y = f (x), qhov chaw uas x = c (t), ces tus muaj nuj nqi y, nrog rau kev hwm rau cov nce mus nce los t, hu ua lub complex.

Yog li, nyob rau hauv cov zauv tsom xam ntawm lub derivative ntawm ib tug puas muaj nuj nqi yog kho raws li ib tug derivative ntawm cov nuj nqi khoo los ntawm cov derivative ntawm nws cov sub-zog. Rau qhov yooj yim ntawm cov kev cai ntawm ntau yam ntawm txoj kev khiav dej num yog nyob rau hauv daim ntawv ntawm ib lub rooj.

Nrog kev siv cov lus no yog ib qho yooj yim uas yuav tau nco derivatives. Tus so ntawm lub derivatives ntawm complex zog yuav pom, yog hais tias peb thov cov kev cai ntawm ntau yam ntawm kev khiav dej num uas tau raug muab teev tseg nyob rau hauv lub theorems thiab corollaries rau lawv.