Let $\Gamma \subset \mathbf{R}^d$ be a closed, countably $n$-rectifiable set. Is there any reasonable way to write the derivatives $$ \frac{\partial}{\partial x_i} \mathrm{dist}\, (x,\Gamma) $$ for $x \notin \Gamma$?

It is Lipschitz and therefore differentiable almost everywhere. And since $\Gamma$ has tangent planes almost everywhere, it of course has a normal space at almost every point too, so you might hope/guess that for almost every point in some neighbourhood of $\Gamma$, there is a reasonable expression for these derivatives. Is anything like that true?