Question

设z = x² f (x, xy)，其中f 具有二阶连续偏导数，求ð²z/ðxðy,要过程，详细加分

Answer 1

令xy=u，下面在xy为中间变量且要求导时用u表示，即f(x,xy)=f(x,u),ðu/ðx=y,ðu/ðy=x.
求z对x的一阶偏导数ðz/ðx=2xf(x,xy)+x²[ðf(x,u)/ðx+ðf(x,u)/ðu*x)
f 对y的一阶偏导数后面要用到 ðf(x,u)/ðy=ðf(x,u)/ðu*x=xðf(x,u)/ðu
∴ð²z/ðxðy=ð(ðz/ðx)/ðy
=ð(2xf(x,xy)+x²*ðf(x,u)/ðx+x3*ðf(x,u)/ðu)/ðy
=2xðf(x,u)/ðu*x+x²*ð²f(x,u)/ðxðu*x+x立方ð²f(x,u)/ðu²*x
=2x²ðf(x,u)/ðu*x+x立方*ð²f(x,u)/ðxðu+x四次方ð²f(x,u)/ðu²