《高数双语》课件section 2-3

Section2.3Higher–orderDerivativesHigher-orderderivatives2secondorderderivativeHigher-orderderivatives3Definition

(Higherorderderivatives)Supposethatthefunctionisderivable.Ifitsderivativeisalsoderivableat,thenfiscalledtwicederivableatxandthederivateofatxiscalledthesecondderivativeoff

atx,anddenotedby.IffiseverywheretwicederivableonI,thenfissaidtobetwicederivableonI

andiscalledsecondderivativeorsecondderivedfunctionoffonI.Ingeneral,ifisderivableat,thederivativeoforder,thenfiscalledntimesderivableatxandthederivativeofatxiscalledthederivativeofordernoff

atx,andisdenotedby.Higher-orderderivatives4wecandefinederivabilityofordern

onIandthederivativeofSimilarly,ordernonIforthefunctionf.

Definition

(Higherorderderivatives)Thederivativeofordernofthefunctionisdenotedbyor.IfiscontinuousonI,thenfissaidtobecontinuouslyderivableofordernonI,orafunctionofclassonI,denotedby.IffisafunctionofclassonIforanyonI,denotedby,thenfissaidtobeinfinitelyderivableonI,orafunctionofclass.Higher-orderderivatives5Definition(Higherorderderivatives)Thesecondderivativeandderivativesoforderhigherthatsecondorderareallcalledhigherorderderivatives[高阶导数].Commonly,f'issaidtobethefirstorderderivativeoff,andfitselfissaidtobethezeroorderderivativeoff.Higher-orderderivatives6Example

Ify=sin2x,findandSolutionFinish.

Higher-orderderivatives7Example

Provethefollowingformulaeforderivativesofordern:(1)(2)(3)(4)(5)Higher-orderderivatives8Bymeansofmathematicalinductionweknowthattheformula(2)holds.Proof

Wewillprovetheformulae(2)and(5).(2)BecauseAssumethatholds,thenHigher-orderderivatives9Theothersolutionislefttoyou.Proof(continued)

Similarlyto(2)bymeansofmathematicalinduction,wecanobtains(5)BecauseHigher-orderderivatives10Theorem

Supposethatthefunctionsuandvarebothderivableofordern.

Thenanduvarealsoderivableofordern,andmoreoverwehave:(1)

Thelinearproperty:(2)

Leibnizformula:Higher-orderderivatives11Example

LetfindSolutionWetakeItiseasytoseeBymeansoftheLeibnizformulaweobtain:Finish.Higher-orderderivatives12Finish.

Example

Supposethat,Ifitdoesnotexist,explain;ifitexists,findit.where.Doesexist?ProofSince,exists.ByLeibnizformula,wehaveItisfollowsthat.Then,bythedefinitionof,wehaveSo,.Higher-