# Derivative of implicite function.

Let consider the equation

We know that the set of solution of this equation is the set of points in plane at distance 1 from the origin, or a circle of radius 1 with center in (0,0).

We want to give a description of this set depending on a single variable instead of two.

This is possible only local ,not for the whole set of solutions.

But the collection of local solutions can give us a complete information about the circle.

Let with and

For in a small neighborhood of we have

Implicit function theorem says that there is a local function f in a neighborhood of a with values in a neighborhood of b with f(a)=b and

This function is derivable so

But as f has values in a neighborhood of we can suppose so we can divide by f(x) and get

We can continue and have second derivative

In this case there are two distinguished functions and both verifying the above properties.

But what about equation

It is possible to write again y=f(x) ?

Answer is given by the same Implicit function theorem but in this case is not so comfortable to use explicit solution.See Folium of Descartes