设f(x,y,z)=xy^2z^3,且z=(x,y)由方程x^2+y^2+Z^2-3xyz=0确定,求αf/αx

Question

设f(x,y,z)=xy^2z^3,且z=(x,y)由方程x^2+y^2+Z^2-3xyz=0确定,求αf/αx

Answer 1

先由f(x,y,z)对x求偏导
αf/αx=y^2*(z^3+x*3z^2*αz/αx)
再由z(x,y)对x求偏导,
即x^2+y^2+z^2-3xyz=0对x求偏导,可得
2x+0+2z*αz/αx-3yz-3xy*αz/αx=0
整理可得
αz/αx=(2x-3yz)/(3xy-2z)
∴αf/αx=y^2*(z^3+x*3z^2*αz/αx)
=y^2*[z^3+x*3z^2*(2x-3yz)/(3xy-2z)]
=y^2*[(3xyz^3-2z^4+6x^2z^2-9xyz^3)/(3xy-2z)]
=2y^2z^2*[(3x^2-3xyz-z^2)/(3xy-2z)]