  Taking a functional approach, we derive a general expression for the gradient
of the Mutual Information (MI) with respect to the system parameters in the
stochastic systems. This expression covers the cases in which the system input
depends on the system parameters. As an application, we consider the K-user
Multiple Access Channels (MAC) with feedback and utilize the obtained results
to explore the behavior of these systems in terms of the MI. Specializing the
results to the additive Gaussian noise MAC, we extend the MI and Minimum Mean
Square Error (MMSE) relationship, i.e., I-MMSE to the K-user Gaussian MAC with
feedback. In this derivation, we show that the gradient of MI can be decomposed
into three distinct parts, where the first part is the MMSE term originated
from noise, and the second and third parts reflect the effects of the
interference and feedback, respectively. Then, considering the capacity
achieving Fourier-Modulated Estimate Correction (F-MEC) strategy of Kramer, we
show how feedback compensates the destructive effects of the users'
interference in the K-user symmetric Gaussian MAC.
