求二元函数s=x*y*sin(x+y)的一阶偏导数和二阶偏导数

求二元函数s=x*y*sin(x+y)的一阶偏导数和二阶偏导数

一阶偏导数
sx=y*sin(x+y)+x*y*cos(x+y)

sy=x*sin(x+y)+x*y*cos(x+y)

二阶偏导数
sxx=2y*cos(x+y)-x*y*sin(x+y)

sxy=syx
=sin(x+y)+y*cos(x+y)+x*cos(x+y)-x*y*sin(x+y)
=(1-x*y)*sin(x+y)+(x+y)*cos(x+y)

syy=2x*cos(x+y)-x*y*sin(x+y)

s=xysin(x+y),
∂s/∂x=y[sin(x+y)+xcos(x+y)],
∂s/∂y=x[sin(x+y)+ycos(x+y)],
∂^2s/∂x^2=y[2cos(x+y)-xsin(x+y)],
∂^2s/∂x∂y=sin(x+y)+xcos(x+y)+y[cos(x+y)-xsin(x+y)]
=(1-xy)(sin(x+y)+(x+y)cos(x+y),
∂^2s/∂y^2=x[2cos(x+y)-ysin(x+y)].