If z=1/(x^2+y^2-1)

Question

is x(dz/dx)+y(dz/dy)=-2z(1+z) or -z^2(1+z)
note:dz/dx=partial derivative of x
and dz/dy=partial derivative of y??

Answer 1

∂z/∂x = -2x/(x^2+y^2-1)^2
∂z/∂y = -2y/(x^2+y^2-1)^2

so,

x ∂z/dx + y ∂z/dy = -2(x^2+y^2)/(x^2+y^2-1)^2
= -2z^2(1+z)