设Z=f(y/x,y),f有二阶连续偏导数,求az/ax,az/ay,az/axay,

令u=y/x v=y
z=f(u,v)
az/ax=af/au*au/ax+af/av*av/ax=af/au*(-y/x^2)
az/ay=af/au*au/ay+af/av*av/ay=af/au*(1/x)+af/av
a^2z/axay=a(az/ax)/ay=af/au*(-1/x^2)+[a^2f/au^2*au/ay+a^2f/auav*av/ay]*(-y/x^2)
=af/au*(-1/x^2)+[a^2f/au^2*(1/x)+a^2f/auav]*(-y/x^2)
=af/au*(-1/x^2)+a^2f/au^2*(-y/x^3)+a^2f/auav*(-y/x^2)
再问： 　　a^2z/ay^2=?
再答： a^2z/ax^2=a(az/ax)/ax=af/au*(2y/x^3)+a^2f/au^2*(y^2/x^4) a^2z/ay^2=a(az/ay)/ay=(a^2f/au^2*(1/x)+a^2f/auav)*(1/x)+a^2f/auav*(1/x)+a^2f/av^2 =a^2f/au^2*(1/x^2)+a^2f/auav*(2/x)+a^2f/av^2