三角函数二阶偏导数练习题I5.docx

T1：y=sin"x求二阶导数。T2：求函数y-cos9xtan8x的二阶导数。4C上十业/COS(5x+30),乙一7人n业,心：求函数y=-的-阶导数。丁4：求Z=Sin(X'+3Oy)的二阶偏导数。T5：求z=sin3(6x+10y)的二阶偏导数。守6：求函数z=si%rrV+e'的二阶偏导数。T1：y=sin"求二阶导数。解：y=si6,求复合函数求导法则有：y,=6sinx*cosx;y,=6(5sin4xcos2-sin5xsinx),=6sin4x(5cos2x-sin2x)oT2：求函数y=cos9xtan8x的二阶导数。解：y=cos9xtan8x,求函数乘积求导法则有：y,=-9sin9xtan8x+8cos9xsec28×y,=-9(9cos9xtan8x+8sin9xsec28x)+8(-9sin9xsec28x+16cos9xsec28xtan8x),=-92cos9xtan8x-144sin9xsec28x+128cos9xsec28xtan8×o4c上十业,COS(5x+30),乙一7人n业,T3：求函数尸的一阶导数。X&Itcos(5x+30)上？九*上曰.j,n解：y=,求函数商求导法则有：-5sin(5x+30)-cos(5x+30)y=；5sin(5x+30)x+cos(5x+30)A=5=x2B,11a,b-ab,y=一仅一,其中：A'=52cos(5x+30)x+5sin(5x+30)-5sin(5x+30)=52cos(5x+30)x,B'=2x,代入上述二阶导数，有：52cos(5x+30)x*x2-2x5sin(5x+30)x+cos(5x+30)：*4i52cos(5x+30)x2-25sin(5x+30)x+cos(5x+30)3oT4：求z=sin(x2+30y)的二阶偏导数。解：z=sin(x2+30y),由全微分求导有：dz=cos(x2+30y)(2xdx+30dy),=2xcos(x2+30y),=30cos(x2+30y);继续求二阶偏导数有:cos(x2+30y)-2xsin(x2+30y)x;相/-2=-302sin(x2+30y);oya：=A：=-2XSin(x2+30y)*30=-60xsin(x2+30y)oxyyoxzTs5:求z=sin3(6x+10y)的二阶偏导数。解：Z=Sin3(6x+1Oy),由偏导数求导原则有:z。妆18sin2(6x+1Oy)cos(6x+1Oy),再对x,y求二阶偏导数有:°2t3*62sin(6x+1Oy)2cos2(6x+1Oy)-sin2(6x+1Oy),Vz=30*1Osin(6x+1Oy)2cos2(6x+1Oy)-sin2(6x+1Oy);oy2°2a=aa=18*1Osin(6x+1Oy)2cos2(6x+1Oy)-sin2(6x+1Oy)0xyyoxW求函数Z=Si丹-的二阶偏导数。解:即:z=sin-4y+e4,由全微分求导公式有:Xdz="11cosd-4x3yd-6x4y5dy,z11136T-=-I1cos_*-2-4xy,xXXy，2Z2his®=-6x4y5,进一步求二阶偏导数为:I11CC11136-*1+22COS-*-3-12xy,XXXX2Z2Z35dxdy=dydx="24xy"