zxandzy as functions of x y and z assuming that the function

?z?xand?z?y as functions of x, y, and z, assuming that the function z=f(x,y) is given implicitly by the equation x^3 + xyz = yz.

differentiate with respect to x:

d/dx (x^3 + xyz = yz)

3x^2 + y*z + (xy)(dz/dx) = y(dz/dx)

(dz/dx)(y-xy) = 3x^2 + yz

thus z/x = (3x^2 + yz)/(y-xy)

now differentiate with respect to y:

d/dy (x^3 + xyz = yz)

0 + x*z + (xy)(dz/dy) = y(dz/dx) + 1*z

(dz/dx)(y-xy) = xz-z

thus z/y = (xz-z)/(y-xy) = z(x-1)/y(1-x) = -z/y