Maximum likelihood estimator for multiplicative Gaussian noise
OK, let's have a look at one of the problematic terms:
$$
\frac{\delta}{\delta x} \bigg[ \bigg(\frac{\tilde{d}[n]}{x}-\mu_A\bigg)^2 \bigg ] 
= - \frac{2 \tilde{d}[n] \bigg (\tilde{d}[n] - \mu_A x\bigg) }{x^3}
$$
which can be verified by Wolfram Alpha.

The full derivative of the summation term is then just this summed over $n$.