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I have to solve a system f(x,y,z)=0 in the neighbourhood of (1,1,1). I have a problem computing the derivative of an implicit function (x,y)=g(z), whose existence is given by the implicit function theorem when applied to the given function f(x,y,z) which goes from R^3 to R^2. I use as it seems two equivalent standard methods.

I compute the Jacobi matrix of f and check whether the minor - a square matrix - is invertible at the given point. Then I invert it and with one more step get the derivative of g at z=1.

The other method is: I use implicit differentiation, i.e. I differentiate the system f(x,y,z)=0 directly and get differentials dg_1/dz, dg_2/dz (because g is two-component).

Now I get two different answers with either method. The difference is only the sign, but I cannot figure out WHY there is sign change! Is there anything I should pay attention to when I use these two methods? I mean they are basically the same! I checked each step in both of them. What could be the reason for the difference?