Derivative of a mutual information for a Gaussian channel

Question

I saw this question which inspired me to ask mine. Given that I have a mutual information $$ I\left(X;\sqrt{\frac{1}{(x/\delta)+\sigma^2}}\cdot X + Y\right), $$ which is a function of $x$ that is not a random variable, and $\delta$ and $\sigma$ are fixed constants that are not random variables either. Furthermore, $X\sim\text{Bernoulli}(\nu)$ and $Y\sim N(0,1)$ are independent random variables -- note that $\nu$ here is also some fixed constant. So mathematically, I am asking what does the following evaluate to? $$ \frac{\partial}{\partial x}I\left(X;\sqrt{\frac{1}{(x/\delta)+\sigma^2}}\cdot X + Y\right). $$ Edit: I believe we can overcome the tedious manipulations with the results from equation (2) of this paper.