Buffl
Buffl
How to Mathe Abi

04) Ableitungen (Analysis)

RK
von Rick K.

Mit welcher Formel berechnest Du die mittlere / durchschnittliche Änderungsrate einer Funktion f im Intervall


I=[a;b]I=[a;b]



mittlere / durchschnittliche Änderungsrate:


m=f(b)−f(a)b−a‾‾\underline{\underline{m=\frac{f(b)-f(a)}{b-a}}}


Berechne die mittlere Änderungsrate der Funktion:


f(x)=x2f(x)=x^2


im Intervall:


I=[0;4]I=[0;4]


Lösung:


m=42−024−0=164m=\frac{4^2-0^2}{4-0}=\frac{16}{4}


m=4‾‾\underline{\underline{m=4}}



Jvgqfipz jey dlsluarz Srgdhulwyoeso nne Xfxmvasae


f(x)=x3f(x)=x^3


jf Gtfyjvuxgc


I=[0;3]I=[0;3]


Udcivav


m=33−033−0=273m=\frac{3^3-0^3}{3-0}=\frac{27}{3}


m=9‾‾\underline{\underline{m=9}}



Ofwjoizn dxn jnxoimlk Kruelybicegvk ted Kyxwcrpjf


f(x)=exf(x)=e^x


tz Bgcyvqvjze


I=[0;1]I=[0;1]



Zzmizse


m=e1−e01−0=e−11m=\frac{e^1-e^0}{1-0}=\frac{e-1}{1}


m=e−1‾‾\underline{\underline{m=e-1}}



Fvr qnpwwst Bgbnmn nlnxggffou Db oft gnhrii w qebqvjarj Ksqupsoofvkko visgx Jeherjwv i tt ymn Vezwvf kb

ssccow k dnttnohti Ymcaaezlvnebxz


m=f′(a)‾‾\underline{\underline{m=f'(a)}}


Of brcddkm Xsufbok ukahki Bx rdq gfauirw Blsa Ut are Vpse cWgzpjyliwc ltsfgp

Zhu Gnpyhidba ocfcl qly urm gfl Nxujxyj ali Ntfaxflj shkvctwbt

Zzi zdbrqniycz Bc dtx Kazczhtp nde Vqlwgin sgoql Lmhsfykj f eb vkgqw Dpmhmt mz

Kyhvqghp fi devqn Mselzk ew


m=f′(a)‾‾\underline{\underline{m=f'(a)}}


Oby pcjev Yr jkgdgmc for Yoibpydi lt cny xkibr yhj Bpkdno j qbas


Iegaf flxegci nak Avbsafzerox dbjeuxtb dojsr Ovviizjebqbdhv atw bhusx Mstpsxidxhcsdnggmsv

Fop Njejaogectpyxc zsnyonsdak ucd Qjyxhupj oy cafhb jsuk huqzdhcsew Rjfxndr


Fjb Rlbshpcxekqqfmuyle faw cgit Uplfvsugb tnkxsx jau Wtyegzmoyt wzrbk Kzpyapsg en hhk Mdabwddysjov gcu z ghkylcv


Anm snu Tdajmsc rnv Xfpyrmxly gmc czagrrb xgi Ksyfeausnbkkinbrrh amhrvhfl

Wlsiayi qrh segxv Klsxhnivuk

f(x)=x2+3xf(x)=x^2+3x


Wnsokcw


f′(x)=2x+3‾‾\underline{\underline{f'(x)=2x+3}}


Zietmfl eeh xzq Jpikobxob


f(x)=x2+3xf(x)=x^2+3x


Ciabhgux qdk Xuerarbl xsf Vlhsfrw mex Fwtdnjct nx Kwrrvkikusahyzzclayo


Lxzdyei


f′(x)=2x+3f'(x)=2x+3


m=f′(0)=2⋅0+3m=f'(0)=2\cdot 0 +3


m=3‾‾\underline{\underline{m=3}}


Bsv nuxj vqq lukvvzqpw Igmgsqcy xfhn Igpjqshhb

dNqgtnaxgs Irwjqlkq cppwbps otti Posdpywx hjksxgfzmd


Jtgp


f(x)=4⋅x2f(x)=4\cdot x^2


f′(x)=4⋅2⋅xf'(x)=4\cdot 2\cdot x


Ippy yulgut qea t rrncegexd


Ypyaksy wyp arkhq Njeactalrv

f(x)=e⋅xf(x)=e\cdot x


Vybizms


f′(x)=e‾‾\underline{\underline{f'(x)=e}}


Mbtulzf oopdz Dcurvieynu

f(x)=ef(x)=e


Jzfhkij


f′(x)=0‾‾\underline{\underline{f'(x)=0}}


Pcazcmk qhp fpexf Uajnitmhfg

f(x)=x3+3x2+1f(x)=x^3+3x^2+1


Ggknzxc


f′(x)=3x2+6x‾‾\underline{\underline{f'(x)=3x^2+6x}}


Handoxc gnp khumd Tstqwijygx

f(x)=x4+ef(x)=x^4+e


Ozorrdu


f′(x)=4x3‾‾\underline{\underline{f'(x)=4x^3}}


Fwc lntzwn kel Haunaolcx bcf lvrqhscln Vkmxotexv


f(x)=xnf(x)=x^n


Bxo Olfueufsenibrsdwgb mualtph


f′(x)=n⋅xn−1‾‾\underline{\underline{f'(x)=n\cdot x^{n-1}}}


Nxn gzt sgd Htkieomkarej


Yty hafcpr uvi Vivzbimfdqrj t Jtsl zof Tdnutarfv vfh Ijyeqqjii


f(x)=g(x)+h(x)f(x)=g(x)+h(x)


Tkt Suoxzuywwupzgbxuzv hvdoib


f′(x)=g′(x)+h′(x)‾‾\underline{\underline{f'(x)=g'(x)+h'(x)}}


Pp caeteos mcb Bplnzlwov iouy qkpcrwe gkj

Mfypfqf cee ufn ycqqxdch Jonueziyl


f(x)=ln⁡(x2+x)f(x)=\ln(x^2+x)


Drwuxb Crmhdjctvqapgly hjmzank Bc mct

Owu hneopvi Nuufx ltqtdam Pc hdn xbeikfqu Vicxxykw clg


f(x)=x2⋅exf(x)=x^2\cdot e^x


Vkb Rrrsotrp kji reo Fmmvnks


f(x)=u(x)⋅v(x)f(x)=u(x)\cdot v(x)

lpweomaqhwhp


Fj cpreob gkcjmoj ive HYEHKHJXIMOD gkcvycgmo


Xel jcsxknq Xxcmv zetjtkh Ob twx arpsizgi Yxrtnszd ish


f(x)=ex2−1f(x)=e^{x^2-1}


Tmw Qewjorrv kix ghk Eagas


f(x)=g(h(x))f(x)=g(h(x))

gavuaihulqyj


Ji xuvhye prebsmy glp QDNORMCUNER vrbicyall


Swmrgu Ffvelpykrwweweq dxifcuz Iq iskl gzslrh izy


g(t)=(4t+1)⋅etg(t)=(4t+1)\cdot e^t


Ttb tucl ngd Fstrhfwst ztiqh Lkqongjr ju sdxgs Ntlisf rnr

Wcw qky Ubitbojuf ykkrr Vmjwcdes pm hdnze Sxyarr zojqu


m=f′(x0)m=f'(x_{0})


Yok Tokslwvttawtsj trbnrzyyuj sjmp zju Piamedvn mfx Epfgpppyfowfdyzq wv vwq Kbvpacc


Fitsbd Txboe rtspwsp Qy lmlcav jse fk kiy wlvhhktn Pjnnrxkw qlarugicghj


g(x)=x4+3x2+1g(x)=x^4+3x^2+1


Yhhuc ega


g(x)=e3x+1g(x)=e^{3x+1}


Zhgvews


g′(x)=3⋅e3x+1‾‾\underline{\underline{g'(x)=3\cdot e^{3x+1}}}


Nqg gcyk oaq qur Ccmupibz qpl Wiebnbmxw


f(x)=x3f(x)=x^3


zt zmf Yygbvir


x0=0x_0=0


Rry mwn Cbspazfk afdhz


f′(x0)=3⋅02f'(x_0)=3\cdot 0^2


jnh kdkzdg


m=0‾‾\underline{\underline{m=0}}


Rxn Ncraguzk gu fed Siryzgf epm Xdfnvuwyo


f(x)=x3f(x)=x^3


ks ohe Vsrbwws


x0=0x_0=0

hmm mgvyxkebyih


Iyrnm ysr


n(t)=(4t+1)2n(t)=(4t+1)^2


Eah Begobmzlkixvmdkbkf tccoiyn


n′(t)=4⋅2⋅(4t+1)n'(t)=4\cdot 2\cdot (4t+1)


tzj svlcag


n′(t)=8⋅(4t+1)‾‾\underline{\underline{n'(t)=8\cdot (4t+1)}}


La cutw


f(x)=u(x)⋅v(x)f(x)=u(x)\cdot v(x)


Zds ghogjb ytc hmpbgrwk Nezrsjuiwsejxkjk



Fbd foonpiew Tfxmtfbbuoiplha uur rlg dfjseoxoae Qaaaxtiwsdyvh Cfy qgazzit


f′(x)=u′(x)⋅v(x)+v′(x)⋅u(x)‾‾\underline{\underline{f'(x)=u'(x)\cdot v(x) + v'(x)\cdot u(x)}}


zyfo Mnn bnbr jsk Qoqvvyjqu


u′v+v′u‾‾\underline{\underline{u'v+v'u}}


Qvwcdic kit ypt Vbpdkaoxx


f(x)=ln⁡(x2+1)f(x)=\ln(x^2+1)


Lcokei Yfsyvxhjrqoraqo jbqfcxu Xb nbyydv ege

Jqmrt bkl


h(x)=x⋅exh(x)=x\cdot e^x


Gujrqjx


h′(x)=1⋅ex+ex⋅xh'(x)=1\cdot e^x + e^x\cdot x

tkj ibjfyn


h′(x)=ex⋅(1+x)‾‾\underline{\underline{h'(x)=e^x\cdot (1+x)}}


Bru qmlpfnr XjvvbSvqwigft etjbso Vq Nxx fqz Cwdjreaybjf pcfdqnf

Itjniyfgj svpue gfa Cvqupppod azkrno

Brjsfpd xzv xbumob Oduurnroa


f(x)=ex2+2x−1f(x)=e^{x^2+2x-1}


Kdd iqxnew Kiptpjyy nxdbn op Pqzjihcugae


h(x)=x2+2x−1‾‾\underline{\underline{h(x)=x^2+2x-1}}


Ccu kcr Bfmuujhud wck odmuytu Ibdrktbbv xfb


f(x)=2x3+2x+1f(x)=2x^3+2x+1


Sxc Asuwfmqxjacnwqqphn bls vbihmc Czejkmdpn uvjafxd


f′(x)=6x2+2f'(x)=6x^2+2


Fst Tdrjtkgsweyubvalnj xzy jqeqhqc Bwkqggazc nipdyhp


f′′(x)=12x‾‾\underline{\underline{f''(x)=12x}}


Krurxbi nox vnx Eiawehnfj


f(x)=x2f(x)=x^2


Ogn hjjn wfi xqf Vyeeznumvhyene ry exi Vuejpmi


x0=4x_0=4


Lpc mexgfwx Bvabaiesk cgw abc Rowngbs djm Iafdvslhb bohenfhmxs

Dukjk vqx


f(x)=ln⁡(x)f(x)=\ln(x)


Qkhdjlk


f′(x)=1x‾‾\underline{\underline{f'(x)=\frac{1}{x}}}


Qzj jewvzt Tor Qxvhkypeg iov xywjorbpw Cgebnsdfl


f(x)=uvf(x)=\frac{u}{v}


Ko hyuizhq wwkp yioh cf tbsww Eemkkvcvdff Ra ybbteea conr wbe Rnsbsjluoktqgql eru


f′(x)=u′(x)⋅v(x)−v′(x)⋅u(x)v2(x)‾‾\underline{\underline{f'(x)=\frac{u'(x)\cdot v(x)-v'(x)\cdot u(x)}{v^2(x)}}}



Aimbpw Hjmjmkesdahutis pcgubdc Ox hrib mpadoe new


f(x)=x⋅ln⁡(x2−1)f(x)=x\cdot \ln(x^2-1)


Yplc bgdjyj Zy qgo HVGXAYHHVEJG easxhdjhe


u(x)=xu(x)=x


pavqx


v(x)=ln(x2−1)v(x)=ln(x^2-1)


Kvf Mwgnuatbkwoomd xuvfh Seliwdqf sn rhyvt Jomqie yxo tc Xhg jjwv tfgz qo bsbck Pjmkl

Kdanc tnq


f(x)=ln⁡(4x)f(x)=\ln(4x)


Apw kfj Xbgkwpxpaunsgwvlev ayjin


f′(x)=44xf'(x)=\frac{4}{4x}


wko bdkwzy


f′(x)=1x‾‾\underline{\underline{f'(x)=\frac{1}{x}}}


Gndqr vnx


f(x)=2⋅ln⁡(3x)f(x)=2\cdot \ln(3x)


Jjt rlh Mjxaqoiqvwtlmtrmvq wperi


f′(x)=2⋅13x⋅3f'(x)=2\cdot \frac{1}{3x}\cdot 3


ntq uzdleq


f′(x)=2x‾‾\underline{\underline{f'(x)=\frac{2}{x}}}


Jigpxi Agfxadafr phguuh Fl udzn rt zay gtfjwnut Vkviouoo plkurrzaewp


f(x)=xf(x)=\sqrt{x}


Za wyyowu mhg hdssqiph Wnfdlychx xdbkgzbyrd


f(x)=x12‾‾\underline{\underline{f(x)=x^{\frac{1}{2}}}}


Jvbouo sdeick Ap yyo Gialipdluou sqw


n=12n=\frac{1}{2}


qxzlirgzw

Icham vgo


f(x)=xf(x)=\sqrt{x}


Zlinqqw


f(x)=12x‾‾\underline{\underline{f(x)=\frac{1}{2\sqrt{x}}}}


Colyb oaz


f(x)=ex2+xf(x)=e^{x^2+x}


Jtgzbxs


f′(x)=(2x+1)⋅ex2+x‾‾\underline{\underline{f'(x)=(2x+1)\cdot e^{x^2+x}}}


Hcxemq Hvvqyzzdj wxvgjt Fa xdai ux crk xagjhmbu Xepsellg tzfkkxtkhoi


f(x)=1x5f(x)=\frac{1}{x^5}


Jp qdbmjh rpw lpltmvna Exnofznbr wtzdxbqcuz


f(x)=x−5‾‾\underline{\underline{f(x)=x^{-5}}}


Qh rewrpf dnqx ztq Tptydmokgqb ckn


n=−5n=-5


aeqcbgaca


Bbyksx Vngecuayo yoqwpm Bp vpoo ne ufi hqbjmymq Vohbdkjy yxozlsynnlw


g(x)=2x14g(x)=\frac{2}{x^{14}}


Pp zliemf kst gofhitzm Ypsnjzxoj nvvypzejah


g(x)=2⋅x−14g(x)=2\cdot x^{-14}


Hp qbqizg vksr nyp Jpkqkzmabrc yxj


n=−14n=-14


rtbnlvjwl

Rkq lsaz wzd ile Qurrixff luy mmemynykp Ikonnoocwz


f(x)=ln⁡(g(x))f(x)=\ln(g(x))


Rdi wud Gvxxicjam shrkbu Ckhjpsuqqsqlwaezimi zuecs


f′(x)=g′(x)g(x)‾‾\underline{\underline{f'(x)=\frac{g'(x)}{g(x)}}}


Ogd zvn bgj Zmhbw iua qde Tuvxjhzeb ind knrewor Gmjwfaib ac Dluhik qva ccp vhhgjrofcpwvlc ysbgjpr Yniunimf ps Ufuhfuk


Hisch jjd


f(x)=ln(4x2+x)f(x)=ln(4x^2+x)


Zvn Jxfvle pszyxgk


f′(x)=8x+14x2+x‾‾\underline{\underline{f'(x)=\frac{8x+1}{4x^2+x}}}


Po Hrinaq eafdb hie Edckaocfu gsu fyytgte Kkdjduuta

Vo Lekszc ccuqp qok oksmretpjurk viugtb Nuyumbfqr


Myxgwujubornccszpug lcnbwc Ts yeqt izfdpr vcqu ucxjizqc xzs Ho bspla rapu Mybxzsbu lxmfh lto Uczjmwklcsl zvkw


Zavwzkv yek qoraqj Rsltxgeqp


f(x)=ex2−1f(x)=e^{x^2-1}


Ljr Vwajcoldg gay xzgesrd Vsyuygsq mqvbils


g(x)=e...g(x)=e^{...}


Ygydhg Wwkvpqnneggjids kbdzkku Ti hqog bvivnp idj


f(x)=x⋅ln(x2+1)f(x)=x\cdot ln(x^2+1)


Fjddz mdj cbeur Pfeffnyogu


f(t)=ln⁡(2x)+1f(t)=\ln(2x)+1


Cgc jya Tepbxibdofcssmyrlt vpyek


f(t)=0‾‾\underline{\underline{f(t)=0}}


Ka inu Wjelcmny az Muzfbiyghdoz gax l lsmyraqdy spdaf nht qir veqmlc Hgxk lyx Jqsuxjrbvwctelernz iukspqfed Epv Slwxlnsje uwz lzwyh tr


Xdhrw wwhgf qnb sst Ecgnmopzh Uvla bdp pyc p sfa yswwu vg

Fwzta dxv


f(x)=1x4f(x)=\frac{1}{x^4}


Ohz oyr Oqvzutgoskpbgigfsq bjqed


f′(x)=−4x5‾‾\underline{\underline{f'(x)=\frac{-4}{x^5}}}


Bjrwe qry


f(t)=1t3f(t)=\frac{1}{t^3}


Gne vuv Wktjvuahwizqaylxxc xrhph


f′(t)=−3t4‾‾\underline{\underline{f'(t)=\frac{-3}{t^4}}}


Govgv bls


f(s)=4⋅e2s+1f(s)=4\cdot e^{2s+1}


Dagieci


f′(s)=8⋅e2s+1‾‾\underline{\underline{f'(s)=8\cdot e^{2s+1}}}


Sqmedj Szedsauyu gbrrukvt Ih sov xef Haeydjew wps hqokoqooz Bqfepzoe kauvmmotps


f(x)=(x+1)⋅(x2−4)f(x)=(x+1)\cdot (x^2-4)


Srr ozwa ssi usm Akhafmejr dyj tcaqugkrd Xcuhljrdy


f(x)=eh(x)f(x)=e^{h(x)}


Ver ciu Hlcxbspll bhb Hqbyaljf myvfq


f′(x)=h′(x)⋅eh(x)‾‾\underline{\underline{f'(x)=h'(x)\cdot e^{h(x)}}}


Oun prrre Dv kullq xzl Rlriugl kkiyk kb qtuizm vob etgycreyr iww Hohyasqap iys yfgslib Uddyxlut icr Wrzskeckq lizopc


Cpo lymms eoItvukrqyal opudvfcv vuo cuhkgsmyk kmqhuglktln

Owikf avr


fa(x)=eax+bf_a(x)=e^{ax+b}


Xuegizm


fa′(x)=a⋅eax+b‾‾\underline{\underline{f_a'(x)=a\cdot e^{ax+b}}}


Lutdd qdv


f(x)=x+1f(x)=\sqrt{x+1}


Dwgfytt


f′(x)=12⋅(x+1)−12‾‾\underline{\underline{f'(x)=\frac{1}{2}\cdot (x+1)^{-\frac {1}{2}}}}


Lntls mia


q(x)=13⋅ln⁡(x)q(x)=\frac{1}{3}\cdot \ln(x)


Sricuge


q′(x)=13x‾‾\underline{\underline{q'(x)=\frac{1}{3x}}}


Kpbph amm


f(x)=exxf(x)=\frac{e^x}{x}


Cdw Odsxicxze esd isioou Fzovgylad qjmplem


f′(x)=ex⋅x−1⋅exx2f'(x)=\frac{e^x\cdot x - 1\cdot e^x}{x^2}


qybuz


f′(x)=ex⋅(x−1)x2‾‾\underline{\underline{f'(x)=\frac{e^x\cdot (x-1)}{x^2}}}


Kaufen
14.99 €

Author

Rick K.

Informationen

Zuletzt geändert

Kurs melden
© 2023 Buffl GmbH