三角函数二阶偏导数练习题I9.docx

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1、T1：y=sinx求二阶导数。T2：求函数Y=Cosxtan1Ox的二阶导数。八八十业/cos(3x30),.”人D业,T3：求函数尸的二阶导数。XT4：求Z=Sin(X+28y)的二阶偏导数。T5：求Z=Sin1(4x+42y)的二阶偏导数。T6：求函数Z=Sin-2y2+e5的二阶偏导数。XT1：y=sinx求二阶导数。解：y=sin4x,求复合函数求导法则有：y,=4sin3x*cosx;y,=4(3sin2xcos2-sin3xsinx),=4sin2x(3cos2-sin2x)oT2：求函数Y=Cosxtan1Ox的二阶导数。解：y=cosxtan1Ox,求函数乘积求导法则有：y=-

2、sinxtan1x+1Ocosxsec2IOx;y,=-(cosxtan10x+1Osinxsec210x)+10(-sinxsec210x+20cosxsec210xtan10x),=-cosxtan10-20sinxsec210x+200cosxsec210xtan10o4c上十业,COS(3x+30),乙一7人n业,T3：求函数尸的一阶导数。X&Itcos(3x+30)上？以*上曰7一解：y=,求函数商求导法则有：-3sin(3x+30)-cos(3x30)y=；3sin(3x+30)x+cos(3x+30)A=5x2B,11a,b-ab,y=一仅一,其中：A=32cos(3x+30)x

3、+3sin(3x+30)-3sin(3x+30)=32cos(3x+30)x,B=2x,代入上述二阶导数，有:1132cos(3x+30)x*x2-2x3sin(3x+30)x+cos(3x+30)y=-3?cos(3x+30)2-23sin(3x+30)x+cos(3x+30)x3zT4:求z=sin(x4+28y)的二阶偏导数。解：z=sin(x4+28y),由全微分求导有：dz=cos(x4+28y)(4x3dx+28dy),=4x3cos(x4+28y),=28cos(x4+28y)继续求二阶偏导数有:3x2cos(x4+28y)-4x3sin(x4+28y)x3,=4x23cos(x

4、4+28y)-4x4sin(x4+28y);-2=-282sin(x4+28y);oy22-4xsin(x4+28y)*28=-112x3sin(x4+28y)。xyyxT5:求z=sin,(4x+42y)的二阶偏导数。解：Z=Sin(4x+42y),由偏导数求导原则有：ZQ=16sin(4x+42y)cos(4x+42y),z,=168sin(4x+42y)cos(4x+42y);再对x,y求二阶偏导数有:4*42sin2(4x+42y)3cos2(4x+42y)-sin2(4x+42y),U2T168*42sin2(4x+42y)3cos2(4x+42y)-sin2(4x+42y);oyaa=Aa=16*42sin2(4x+42y)3cos2(4x+42y)-sin2(4x+42y)oxyoyx6:求函数Z=Sin4-打飞5的二阶偏导数。解：z=sin-2y4+e5,由全微分求导公式有:Xdz=-10cosd-2xy4d-4x2y3dy,XX田z1014即：-=10cos*2-2xy,xXX=-4x2y3,进一步求二阶偏导数为:=-102sin20cos-2xy=T22y2,2Z2Z3xy-yx-V