Maximum likelihood estimator for multiplicative Gaussian noise So I'm trying to derive an analytical solution for a MLE that should estimate a static value polluted by multiplicative Gaussian noise.

The vector of measurements $\tilde{\boldsymbol{d}}$ is given as $a[n]x_u$ where $a[n]$ is the $n$-th realization of the random variable $A \propto \mathcal{N}(\mu_A, \sigma_A)$ and $x_u$ is an unknown constant that should be estimated from $\tilde{\boldsymbol{d}}$.

I got as far as the log-likelihood function, which I now need to maximize with regard to $x$. To do this, I need to take the derivative of the log-likelihood , set it to zero and solve for $x$. However, the summation term at the end is giving me a headache as I can't figure out its derivative.

$\ell(x, \tilde{\boldsymbol{d}}) = -N \cdot \log x - N\cdot \log\sigma_A - \frac{N}{2} \log 2\pi - \frac{1}{2\sigma^2_A}\sum\limits_{n=1}^N\bigg(\frac{\tilde{d}[n]}{x}-\mu_A\bigg)^2$
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 $n$.