  In this paper, we propose generalizations of the de Bruijn's identities based
on extensions of the Shannon entropy, Fisher information and their associated
divergences or relative measures. The foundation of these generalizations are
the $\phi$-entropies and divergences of the Csisz\'a's class (or Salicr\'u's
class) considered within a multidimensional context, included the
monodimensional case, and for several type of noisy channels characterized by a
more general probability distribution beyond the well-known Gaussian noise. It
is found that the gradient and/or the hessian of these entropies or divergences
with respect to the noise parameters give naturally rise to generalized
versions of the Fisher information or divergence, which are named as the
$\phi$-Fisher information (divergence). The obtained identities can be viewed
as further extensions of the classical de Bruijn's identity. Analogously, it is
shown that a similar relation holds between the $\phi$-divergence and a
extended mean-square error, named $\phi$-mean square error, for the Gaussian
channel.
