复合函数的高阶导数公式及其应用

第9卷第12期南阳师范学院学报Vo.l9No122010年12月JournalofNanyangNormalUniversityDec.2010:2010-08-15:(1990-),,.陈世哲1,陈仕洲2(1.中山大学信息科学与技术学院,广东广州510006;2.韩山师范学院数学与信息技术系,广东潮州521041):给出并应用数学归纳法证明了复合函数的高阶导数公式,推广和改进了相关文献的一些结论.最后给出一些应用实例.:复合函数;高阶导数;数学归纳法:O172:A:1671-6132(2010)12-0022-03[1-2],[2],,.,,[3-6],[6]f(a+bx2).,[6],[6].,,.,.y=f(u),u=(x)dydx=f(u)(x),d2ydx2=f(u)2(x)+f(u)(x),d3ydx3=f(3)(u)3(x)+f(u)(3(x)(x))+f(u)(3)(x),d4ydx4=f(4)(u)4(x)+f(3)(u)(62(x)(x))+f(u)(32(x)+4(x)(3)(x))+f(u)(4)(x)dnydxn=ni=1An,i(x)f(i)(u),An,i(x),,,(n).,1y=f(u),u=(x),f,n,dnydxn=ni=1Pn,i(x)i!f(i)(u),Pn,i(x)=i-1s=0(-1)sCsiusdndxn(ui-s).(1)(x+p)x,p,Pn,i(x)=dndpn((x+p)-(x))ip=0.(2).n=1,2.(1)n=k,dkydxk=ki=1Pk,i(x)i!f(i)(u),dk+1ydxk+1=k+1i=11i!ddxPk,i(x)+Pk,i-1(x)(i-1)!dudxf(i)(u),Pk,0=Pk,k+1:=0.12:Csis=iCs-1i-1,ddxPk,i(x)=i-1s=0(-1)sCsiusdk+1dxk+1(ui-s)+i-1s=1(-1)sCsisus-1dudxdkdxk(ui-s)=i-1s=0(-1)sCsiusdk+1dxk+1(ui-s)+i-1s=1(-1)siCs-1i-1us-1dudxdkdxk(ui-s)=i-1s=0(-1)sCsiusdk+1dxk+1(ui-s)-i-2s=0(-1)siCsi-1usdudxdkdxk(ui-s-1)=Pk+1,j-iPk,i-1dudx,ddxPk,i(x)+iPk,i-1dudx=Pk+1,i.dk+1ydxk+1=k+1i=11i!Pk+1,i(x)f(i)(u).(1)n=k+1.(1).(x+p)x,p,dn(x+p)dxn=dn(x+p)dpn.,dndpn((x+p)-(x))i=dndpnis=0(-1)sCsi((x))s((x+p))i-s=is=0(-1)sCsi((x))sdndpn((x+p))i-s.dn((x+p))i-sdpnp=0=dn((x))i-sdxn=dnui-sdxn,(2).2fn,dndxnf(a+bx+cx2)=mi=0n(n-1)(n-2i+1)i!(b+2cx)n-2icif(n-i)(u),,m=n2,u=a+bx+cx2.(3)u=(x)=a+bx+cx2,((x+p)+(x))n-i=pn-i[(b+2cx)+cp]n-i,dndpn((x+p)-(x))n-ip=0=Cin-i(b+2cx)n-2icin!,Pn,n-i(x)(n-i)!=n!(n-i)!Cin-i(b+2cx)n-2ici=n(n-1)(n-2i+1)i!(b+2cx)n-2ici.1dndxnf(a+bx+cx2)=n-1i=0pn,n-i(n-i)!f(n-i)(u)=mi=0n(n-1)(n-2i+1)i!(b+2cx)n-2icif(n-i)(u),m=n2.1fn,dndxnf(x2)=mi=0n(n-1)(n-2i+1)i!(2x)n-2if(n-i)(x2),m=n2.11[6].,,.111+x2(n)(arctanx)(n).f(u)=u-1,u=1+x2.f(n-i)(u)=(-1)n-iui-n-1.2,11+x2(n)=mi=0(-1)n-in(n-1)(n-2i+1)i!(2x)n-2i11+x2n+1-i,239m=n2.(arctanx)(n)=11+x2(n-1)=ti=0(-1)n-1-i(n-1)(n-2)(n-2i)i!(2x)n-1-2i11+x2n-i,t=n-12.2(arcsinx)(n).(arcsinx)=11-x2,(arcsinx)(n)=(1-x2)-12(n-1).f(u)=u-12,u=1-x2,f(n-1-i)(u)=-12-32-2(n-i-1)-12u12-n+i.2,(arcsinx)(n)=(1-x2)-12(n-1)=ti=0(n-1)(n-2)(n-2i)i!(-2x)n-1-2i(-1)if(n-1-i)(u)=ti=0(n-1)(n-2)(n-2i)2ii!xn-1-2i11-x22n-2i-1,t=n-12.[1]HespelC.IteratedderivativesoftheoutputofanonlineardynamicsystemandFaadiBrunoformula[J].Mathematicsandcomputersinsimulation,1996,42(4/6):641-657.[2],.[J].,2005,20(11):1279-1282.[3]TomMApoto.lCalculatinghigherderivativesofinverses[J].TheAmericanMathematicalMonthly,2000,107:738-741.[4]WarrenPJonson.Combinatoricsofhigherderivativesofinverses[J].TheAmericanMathematicalMonthly,2002,109:273-276.[5]TangQiongandLiuLuohua.CalculatingHigherDerivativesofParametersandComposites[J].,2004,24(3):81-83.[6].[J].,2004,21(5):154-156.HigherderivativeformulaofacompoundfunctionanditsapplicationCHENShizhe1,CHENShizhou2(1.SchoolofInformationScienceandTechnology,SunYatsenUniversity,Guangzhou510006,China;2.DepartmentofMathematicsandInformationTechnology,HanshanNormalUniversity,Chaozhou521041,China)Abstract:Ahigherderivativeformulaofacompoundfunctionwasobtainedanditwasprovedbyusingmathematicalinduction.Someresultsaboutreferenceswereimprovedandgeneralied.Finallysomeexamplesweregiventoillustratethetheorems.Keywords:compoundfunction;higherderivative;mathematicalinduction24