  Unveiling a fundamental link between information theory and estimation
theory, the I-MMSE relationship by Guo, Shamai and Verdu~\cite{gu05}, together
with its numerous extensions, has great theoretical significance and various
practical applications. On the other hand, its influences to date have been
restricted to channels without feedback or memory, due to the absence of its
extensions to such channels. In this paper, we propose extensions of the I-MMSE
relationship to discrete-time and continuous-time Gaussian channels with
feedback and/or memory. Our approach is based on a very simple observation,
which can be applied to other scenarios, such as a simple and direct proof of
the classical de Bruijn's identity. This submission corrects the mistakes in
the previous version.
