  We recover the gradient of a scalar conductivity defined on a smooth bounded
open set in $\mathbb{R}^d$ from the Dirichlet to Neumann map arising from the
$p$-Laplace equation. For any boundary point we recover the gradient using
Dirichlet data supported on an arbitrarily small neighbourhood of the boundary
point. We use a Rellich-type identity in the proof. Our results are new when $p
\neq 2$. In the $p = 2$ case boundary determination plays a role in several
methods for recovering the conductivity in the interior.
