  In this paper we deal with the following family of exponential maps
$(f_\lambda:z\mapsto \lambda(e^z-1))_{\lambda\in [1,+\infty)}$. Denoting
$d(\lambda)$ the hyperbolic dimension of $f_\lambda$. It is known that the
function $\lambda\mapsto d(\lambda)$ is real analytic in $(1,+\infty)$, and
that it is continuous in $[1,+\infty)$. In this paper we prove that this map is
C$^1$ on $[1,+\infty)$, with $d'(1^+)=0$. Moreover, depending on the value of
$d(1)$, we give estimates of the speed of convergence towards 0.
