Suppose $S$ is a (regular) compact differentiable surface embedded in $\mathbb{R}^3$ so that tubular neighborhoods exist. Consider the diffeomorphism to one of them:
$$F:S\times(-\epsilon,\epsilon)\rightarrow N_{\epsilon}(S)$$
$$F(p,t)=p+tN_p$$
where $N$ is the Gauss map, that is invertible. I have to calculate the differential of its inverse, however I think I don't fully understand how am I supposed to express it in general. I know the inverse function $F^{-1}$ takes a point in the neighborhood to a point of the space that has for its two first coordinates the orthogonal projection onto the surface and for the third one the oriented distance from said surface. Every time I try to calculate the inverse I get stuck in some circular reasoning (for example I know how to express the distance in function of the projection and vice versa but I cannot explicitly write both of them). Any help is greatly appreciated.