The information divergence of a probability measure $P$ from an exponential family $ℰ$ over a finite set is defined as infimum of the divergences of $P$ from $Q$ subject to $Q\in ℰ$. All directional derivatives of the divergence from $ℰ$ are explicitly found. To this end, behaviour of the conjugate of a log-Laplace transform on the boundary of its domain is analysed. The first order conditions for $P$ to be a maximizer of the divergence from $ℰ$ are presented, including new ones when $P$ is not projectable to $ℰ$.