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1、数学分析实验偏导数与全微分格式意义D[f,x]计算f关于x的偏导数D[f,x1,x2,…]计算f关于x1,x2,…的高阶偏导数D[f,{xi,n}]计算f关于xi的n阶偏导数D[f,x,NonConstants->{y}]计算f关于x的偏导数，y是x的函数Dt[f,x]计算f关于x的全导数Dt[f,x1,x2,…]计算f关于x1,x2,…的全导数Dt[f,x,Constants->{c1,c2,…}]计算f关于x的全导数，其中c1,c2,…为常数Dt[f[x,y]]求f的全微分D[Exp[x+y+z^2],z]D[x^2+y^
2、2,x,y]D[Exp[x+2y],{y,5}]D[x[1]^2+x[2]^2,x[1]]D[g[x^2,y^2],x]D[g[x,y^2],x,x,y]D[x^2+y^2,x,NonConstants->{y}]D[x^2+y[x]^2,x]D[f[Sin[x],y[x^2]],x]Dt[x^2+y^2,x]Dt[x^2+y^2+z^2,x]Dt[x^2+y^2+z^2,x,Constants->{y}]Dt[ArcSin[x/y]]t1={{"D[Exp[x+y+z^2],z]","D[x^2+y^2,x,y]","D[E
3、xp[x+2y],{y,5}]}","D[x[1]^2+x[2]^2,x[1]]","D[g[x^2,y^2],x]","D[g[x,y^2],x,x,y]","D[x^2+y^2,x,NonConstants->{y}]","D[x^2+y[x]^2,x]","D[f[Sin[x],y[x^2]],x]","Dt[x^2+y^2,x]","Dt[x^2+y^2+z^2,x]","Dt[x^2+y^2+z^2,x,Constants->{y}]","Dt[ArcSin[x/y]]"}};t2={{D[Exp[x+y+z^2],
4、z],D[x^2+y^2,x,y],D[Exp[x+2y],{y,5}],D[x[1]^2+x[2]^2,x[1]],D[g[x^2,y^2],x],D[g[x,y^2],x,x,y],D[x^2+y^2,x,NonConstants->{y}],D[x^2+y[x]^2,x],D[f[Sin[x],y[x^2]],x],Dt[x^2+y^2,x],Dt[x^2+y^2+z^2,x],Dt[x^2+y^2+z^2,x,Constants->{y}],Dt[ArcSin[x/y]]}};TableForm[Transpose[J
5、oin[t1,t2]]]隐函数求导例已知x2+y2+z2-4z=0,求.eq=(x^2+y^2+z^2-4z==0);y/:Dt[y,x]=0;deq1=Dt[eq,x]deq2=Dt[eq,x,x]Solve[{deq1,deq2},{Dt[z,x],Dt[z,{x,2}]}]例设xu-yv=0,yu+xv=1,求eq={xu-yv==0,yu+xv==1};y/:Dt[y,x]=0;x/:Dt[x,y]=0;eqx=Dt[eq,x]{jie1}=Solve[eqx,{Dt[u,x],Dt[v,x]}]eqy=Dt[eq,y
6、]{jie2}=Solve[eqy,{Dt[u,y],Dt[v,y]}]uv=Solve[eq,{u,v}];u=u/.uv[[1]];v=v/.uv[[1]];{D[u,x],D[u,y],D[v,x],D[v,y]}//Simplify{Dt[u,x],Dt[u,y],Dt[v,x],Dt[v,y]}/.jie1/.jie2//Simplify多元函数的极值例1设z=x4-8xy+2y2-3,求函数的极值点和极值.z=x^4-8xy+2y^2-3;dzx=D[z,x];dzy=D[z,y];s0=Solve[{dzx==0
7、,dzy==0},{x,y}]dzxx=D[z,x,x];dzxy=D[z,x,y];dzyy=D[z,y,y];L=dzxxdzyy-dzxy^2Module[{a,b,c,d,e},Do[a=L/.s0[[k]];b=dzxx/.s0[[k]];c=z/.s0[[k]];d=x/.s0[[k]];e=y/.s0[[k]];If[a>0,If[b<0,Print["(",d,",",e,")isamaximumpoint.","z=",c],Print["(",d,",",e,")isaminimumpoint.","z="
8、,c]]],{k,1,Length[s0]}]]也可以通过图形来观察极值点与驻点Plot3D[z,{x,-4,4},{y,-6,6},PlotPoints->100,Mesh->False]缩小值域再作观察:small=Plot3D[z,{x,-4,4},{y,-6,6},P
