Lub derivative ntawm lub sine ntawm lub kaum sab xis yog sib npaug zos rau cov cosine ntawm tib lub

Dana yooj yim trigonometry muaj nuj nqi y = Kev txhaum (x), yog differentiable nyob rau txhua kis ntawm tag nrho cov sau. Peb yuav tsum ua pov thawj tias lub derivative ntawm lub sine ntawm kev sib cav yog sib npaug zos rau cov cosine ntawm tib lub, uas yog, '= cos (x).

Lub pov thawj raws li lub ntsiab txhais ntawm ib tug derivative muaj nuj nqi

Peb txhais tau x (arbitrary) nyob rau hauv ib co me me zej zog ntawm ib tug taw tes x Δh _0. Peb yuav qhia rau cov ua haujlwm tus nqi nyob rau hauv nws, thiab nyob rau taw tes x mus nrhiav tus increment ntawm ib tug muab nuj nqi. Yog hais tias Δh - sib cav incremented, tus tshiab sib cav - qhov no x ₀ + Δx = x, tus nqi ntawm no muaj nuj nqi rau ib tug muab tus nqi ntawm lub cav (x) yog sib npaug zos txhaum (x ₀ + Δx), cov nuj nqi muaj nqis nyob rau ib lub point (x ₀₎ yog tseem paub .

Tam sim no peb muaj Δu = Kev txhaum (x ₀ + Δh) -Sin (x ₀₎ - tau increment muaj nuj nqi.

Raws li cov mis ntawm sine sum ntawm ob teej ces kaum peb yuav hloov qhov txawv Δu.

Δu = Kev txhaum (x ₀₎ · cos (Δh) + cos (x ₀₎ · txhaum (Δx) Rho tawm kev txhaum (x ₀₎ = (cos (Δx) -1 ) · txhaum ( x ₀₎ + cos (x ₀₎ · Kev txhaum (Δh).

Tau permutation cov nqe lus grouped thawj rau peb txhaum (x _0), coj tawm cov zoo tshaj - sine - lub nkhaus. Peb tau txais nyob rau hauv lub qhia cos sib txawv (Δh) -1. Nws sab laug mus rau hloov tus kos npe rau hauv pem hauv ntej ntawm lub parenthesis thiab nkhaus. Paub txog dab tsi yog lub 1-cos (Δh), peb ua qhov kev hloov thiab tau yooj yim qhia Δu, uas yog ces muab faib los ntawm Δh.
Δu / Δh yuav muaj rau hauv daim ntawv: cos (x ₀₎ · txhaum (Δh) / Δh 2 · Kev txhaum ² (0.5 x Δh) · txhaum (x ₀₎ / Δh. Qhov no yog tus piv ntawm cov increment ntawm cov nuj nqi rau cov mus kawm rau hauv increment ntawm lub cav.

Nws tseem mus nrhiav tau cov kev txwv ntawm cov ratios tau los ntawm peb lub sij hawm Lim Δh, tsis dim rau xoom.

Nws yog lub npe hu hais tias cov kev txwv kev txhaum (Δh) / Δx yog sib npaug zos rau 1, nyob rau hauv lub mob. Thiab qhov qhia 2 · Kev txhaum ² (0.5 x Δh) / Δh nyob rau hauv lub resulting sum kev transformations rau cov khoom uas muaj raws li thawj d zoo kawg li txwv: lub numerator ntawm lub feem thiab znemenatel faib los ntawm 2, lub xwmfab ntawm lub sine hloov khoom. Ntawm no yog yuav ua li cas:
(Kev ua txhaum (0,5 · Δx) / (0,5 · Δx)) · txhaum (Δx / 2).
Cov kev txwv ntawm no qhia thaum Δh nyhav pes tsawg, yuav ua ncaj rau tus xov tooj ntawm xoom (0 multiplied by 1). Nws hloov tawm hais tias cov kev txwv ntawm tus piv Δy / Δh yog cos (x ₀₎ · 1-0, qhov no yog cos (x _0), cov kev qhia uas yog kev ywj siab ntawm Δh tsis dim mus 0. Qhov xaus: lub derivative ntawm lub sine ntawm tej lub yog sib npaug zos rau x cosine ntawm x, yuav tau sau ntawv raws li: y '= cos (x).

Cov uas ua cov mis mos muaj npe teev nyob rau hauv lub rooj ntawm cov paub derivatives, qhov twg tag nrho cov elementary zog

Nyob rau hauv kev daws teeb meem, nyob qhov twg nws muaj raws li cov derivative ntawm lub sine, koj siv tau cov kev cai ntawm ntau yam thiab npaj txhij-ua qauv ntawm lub rooj. Piv txwv li: nrhiav tau cov derivative ntawm qhov nyuaj ua y = 3 · Kev txhaum (x) -15. Peb siv cov elementary derivation cai tshem tawm txog zauv ceeb rau lub kos npe rau ntawm lub derivative thiab xam cov derivative qhov tooj (uas yog zero). Thov ib sine rooj nqi ntawm cov derivative ntawm lub kaum sab xis x sib npaug zos cos (x). Tau txais cov lus teb: y '= 3 · cos (x) -O. Qhov no derivative, nyob rau hauv lem, yog tseem yog ib qho elementary muaj nuj nqi y = H · cos (x).

Lub derivative ntawm sine squared ntawm kev sib cav

Nyob rau hauv lub xam ntawm cov kev qhia (Kev ua txhaum ² (x)) 'yuav tsum nco ntsoov yuav ua li cas differentiated complex muaj nuj nqi. Yog li ntawd, ² = Kev txhaum (x) - yog ib lub hwj chim muaj nuj nqi raws li sine squared. Nws sib cav kuj yog ib tug trigonometric muaj nuj nqi, ib tug complex sib cav. Cov tshwm sim nyob rau cov ntaub ntawv no yog sib npaug zos rau cov khoom ntawm cov thawj d yog ib tug square ntawm lub complex derivative ntawm lub cav, thiab lub thib ob - tus derivative ntawm lub sine. Ntawm no yog txoj cai rau differentiating ib tug muaj nuj nqi ntawm ib tug muaj nuj nqi: (u (v (x))) 'yog (u (v (x)))' · (v (x)) '. Qhia ntawm v (x) - ib tug complex sib cav (internal muaj nuj nqi). Yog hais tias qhov muab nuj nqi "y sib npaug rau hauv lub sine squared x", ces tus derivative ntawm no puas muaj nuj nqi yog y '= 2 · Kev txhaum (x) · cos (x). Cov khoom los ntawm cov thawj d ob npaug - derivative paub exponential muaj nuj nqi, thiab cos (x) - derivative qhov ntswg complex sib cav ntawm lub quadratic muaj nuj nqi. Zaum kawg tshwm sim muaj peev xwm yuav hloov los ntawm kev siv cov mis ntawm lub trigonometric sine ntawm lub ob lub kaum sab xis. A: Cov derivative yog kev txhaum (2 · x). Qhov no mis yog ib qho yooj yim uas yuav tau nco, nws yog feem ntau siv raws li ib lub rooj.