多元函数的微分学习题

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1、一、计算题81•已矢口z=Jlndy),求竺至.ar，ay給尹®g(f)+y右解:=(加吋)詁〒J—丄oxdx2^lnx+lnyx12兀Jin(兀y)=(1+xy)y85.设z=^s，nvcos.y,求在点(2,兀）处的偏导数.（中等难度）同理竺=—.彷2yjln(xy)82.已知z=sinXy)+co^(xy),求解：因为OZdx=严'cosxcosy,気旻.（中等难度）dxoy解:dzdx=cos(xy)y+2cos(a}9

2、-sin(

3、y所&&&一53

4、(2?r)=esin2cos2cos;r=-es,n2cos2,⑵十-严sin0=0由对称性可知:——=兀[cosdy)—sin(

5、2xy)]•86.已矢Hz=Vxsin-Xdzdz83•已知z=Intan-,求条莎(中等dz1.y4xyy——=—-7=sincos—dx2yjxxx难度）dzVxy—=——cos—=dyxx4x解:dz12兀1——sec~22x=—esc一,87•已知xtandz~5ysec沦Xtan-x2x2xv=-7cscv84•已知z=(l+xy)y9求皐孚.(中等难度)oxoy解：—=y(l+号)zy=y"l+卩)日,OX/(兀y)=ex+ycos(y-x),求字各.(中等难度)dxdy—=[ex+ycos(y-x)]'x=ex+ycos(y-x)+dxex+y[—sin(y—x)]・(-1

6、)=ex^y[cos(y一兀)+sin(y一x)]—=[ex+ycos(y-x)V=ex^ycos(y-x)+dy89•已知"7^歹宀宀（），求0,x2+y2=0ev,v[-sin(y-%)]•!=exy[cos(y-x)-sin(y-x)].dzdz臥'乔（高难度）88.已知f(x,y)=x+(^-l)arcsi解：当F+Fho时dzdz求示示（中等难度）(x2+r)53(F+b)2解：由于H=[x+(J?-l)arcsinJ-J；当x2+y2=0时dz(0,0)]曲2(05,0)70,0)心一>0dz=(jv)；+g-l)arcsinJ；]；dzdx3，(F+y2)20,—=[x+(

7、y-l)arcsinSy=(兀)；・+(yj)；arcsin+(y-l)(arcsinJ{)；1•arcsin—y.XX=arcsin/牡2yjy2VAy-x)AxX3~3(宀尸尸0,兀2+歹2HQVI2".同理得.X2+)，二090•已知M=arcta-H^)z,求dududu八亠丟莎花•（中等难度）解;牛花晌du-z(x-y)z16u{x-yYln(x-y)91.1+(X—y)2x5z1+(x—y)2x已知比du——=zdxdu

8、x2+y28yy2,求篇器,益（中等难度）于是京“心需"Th'Z=Xyx~]]ny+yx丄=yx~](xlny+1)dxdy95.已知z=xsixH-y),求解：由于—=4x3-8xy2,—=4y3-8x2dx8y于是気心曲聲心曲,d2zd2zdx2，dyd*2z茹.（中等难度）空一16小cC*oxoy解：市于y93•已知z=arctan-X、d2zd2z求左'不—=sin(x+y)+xcos(a*+y),—=xcos(a:+y)dxdya2z等.（中等难度）dxdy解：由于于是d2Z—7=cos(x+y)+cos(兀+y)一兀sin(x+y)ftr=(cos(x+y)-sin(x+

9、y),=—xsind+y)dzdz°z=cos(x+y)-xsin(x+y).dxdyd2zd2zdx2，dy（中等难度）于是d2z_2xyd2z_2xydx2(x+>,2)29dy2(x2+y2)296.已知z=xlnrf^y),求d2zdxdy•解：由于d2z_(x2+y2)-2y2_y2-x2dxdy(x2+y2)2(x2+y2)2*dx94•已知z=y",求兽,茫，等.dxdy~oxoy于是dz_xdyx+y（中等难度）律市于&5痣"“dx2x+y(x+y)?(x+y)?'d2z_xdy2(兀+y)282z1x_ydxdyx+y(x+y)2(x+y)2*97.已知z=arc旳i

10、),求d2zd2z,,d2z・（中等难度）dx2oxoy解:由于dzydzxdxJ1-(兀刃2—9彷Jl-g)2于是d2z一y--A屛l-X2y2J1—尢2),2J(1—Ty2)3=严（1+3小+，）七2）z=sin（xy）+0（x，一），99.设2丿有二阶偏求'Z,其中0（w,V）dxdy导数.解ifi«=x,v=—,有y-=ycos(xy)+^+^-52z-x^-=^==-(-2x2y)2J1—Cry)?_疋y•-Uy)2J(l