【题目】求下列函数的高阶微分:(1)设 u(x)=lnx v(x)=e^x 求 d^3(uv) , d^3(u/v)(2)设 u(x)=e^(x/2) v(x)=cos2x ,求 d^3(uv) , d^3(u/v)

【解析】解:(1)d(w)=vdu+udv=e^xd(lnx)+lnxd(e^x)=e^x(1/x+lnx)dx ,d^2(uv)=d[d(uv)]=d[e^x(1/x+lnx)dx] [(六+lnx)d(e)+ed(六+1n)]dx-[e^x(1/x+lnx)+e^x(-1/(x^2)+1/x)]dx^2=e^x(lnx+2/x-1/(x^( d^3(uv)=d[d^2(uv)]=d[e^x(lnx+2/x-1/(x^2))dx^2] =(lnx+2/x-1/(x^2))dx^2⋅d(e^x)+e^xdx^2⋅d(lnx+2/x-1/(x^2)) =e^x(lnx+2/x-1/(x^2))dx^3+e^x(1/x-2/(x^2)+2/(x^2))dx^(3 =e^x(lnx+3/x-3/(x^2)+2/(x^3))dx^3 d()= du-d dn)-nxde)dx-Inx.edxe2r=(1/x-lnx)/(e^x)dx ,d^2(u/v)=d[d(π/ν)]=d[(1/x-lnx)]dx] =(e^xd(1/x-lnx)-(1/x-lnx)d(e^x))/(e^x)dx=rac(-rac1(x^2)-rac1x-rac(1 d^2(u/v)-d[d(u/v)]=d(lnx-2/x-1/(x^2))/dx^2 =rac(e^x(lnx-rac2x-rac1(x^2))'-(lnx-rac2x-rac1(x^2)(e^x)^x)dx^3 -(2/(x^2)+3/(x^2)+3/x-lnx)/dx^3 (2)d(ωv)=d(e^(x/2)cos2x)=cos2xd(e^(x/2))+e^(x/2)d(cos2x) y=1/2e^(x/2)cos2xdx-2e^(x/2)sin2xdx=e^(x/2)(1/2cos2x-2sin2x) d^2(uv)=d[d(uv)]=d[e^(-z)(1/2cos2x-2sin2x)dx] -[(e^(x/2))'(1/2cos2x-2sin2x)+e^(x/2)(1/2cos2x-2sin2x)'dx^2 =[1/2e^(1/2)(1/2cos2x-2sin2x)+e^(x/2)(-sin2x-4cos2x)]dx^2 =e^(x/2)(-2sin2x-(15)/4cos2x)dx^2 d^3(uv)=d[d^2(uv)] [e(-sn2x-csx)]dx=[e^(1/2)⋅1/2(-2sin2x-(15)/4cos2x)+e^(x/2)(-4cos2x+(15)/2sin2x)]dx^3 =e^(x/2)((13)/2sin2x-(47)/8cos2x)dx^3 d()=d(esecx)=(e sc2)"dx3=[(e^(x/2)^n)sex^(2x+3(e^(x/2))^n(sedx))^y+3(e^(x/2))^7(se=[1/8e^(x/2)sec2x+3/2e^(x/2)sec2xtan2x+6e^(x/2)sec2x(1+2x) +8efs2nx(5+6n2)d3=e^(x/2)sec2x(48tan^22x+12tan^22x+(83)/2tan2x+(49)/8)dx^3