Question

设z=f(x^2,g(y/x)),其中f(u,v)具有二阶连续偏导数，g(t)具有二阶导数，求az/ax，a^2z/axay

设z=f(x^2,g(y/x)),其中f(u,v)具有二阶连续偏导数，g(t)具有二阶导数，求az/ax，a^2z/axay

Answer 1

z=f(x^2,g(y/x))
az/ax=f`1(2x)+f`2g`(y/x)(-y/x²)
=2xf`1-y/x²f`2 g`(y/x)
a^2z/axay
=2x[f``11*0+f``12g`(y/x)(1/x)]-{1/x²f`2g`(y/x)+y/x²g`(y/x)[f``12*0+f``22g`(y/x)/x]+y/x²f`2g``(y/x)/x}
=2x[f``12g`(y/x)(1/x)]-{1/x²f`2g`(y/x)+y/x²g`(y/x)[f``22g`(y/x)/x]+y/x²f`2g``(y/x)/x}