  The I-MMSE formula connects two important quantities in information theory
and estimation theory. It states that in a gaussian channel, the derivative of
the mutual information is one-half of the minimum mean-squared error. Higher
derivatives of the mutual information is related to estimation errors of higher
moments, however a general formula is unknown. In this paper, we derive a
general formula for the derivatives of mutual information between inputs and
outputs of multiple Gaussian channels with respect to the signal-to-noise
ratios. The obtained result is remarkably similar to the classic
cumulant-moment relation.
