background: I'm working on a problem that uses the implicit function theorem to show the existence of a solution. I have a continuously differentiable function $f(x,y)=0$ with nonzero Jacobian at a point $(a,b)$. A direct corollary of the implicit function theorem asserts that for sufficiently small changes in $a$, say $a+\delta$ there is a $b+\epsilon$ such that $f(a+\delta,b+\epsilon)=0$.
Question: How do one find such a new solution in a concrete problem?
Example: Let's look at a simple function like $f(x,y)=x^2+y^2-1=0$. $(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$ is a solution, and the Jacobian of $f$ is nonzero at this point. How do you find a solution of the form $(\frac{\sqrt{2}}{2}+0.1,y)$?
Note: I know that the above example can be solved directly, but I want to make use of a method (something like Newton's method maybe) for higher dimensional problems, where $x=(x_1,\ldots,x_m)$, and $y=(y_1,\ldots,y_n)$