What is the first order derivative of the MMSE estimator over the observation?

Question

Consider the observation $y=x+n$, where $n\sim N(0,\sigma^2)$ and the priori distribution $x\sim p_0(x)$. I know the MMSE estimator is given by $\hat{x}=E[x|y]$. What is derivative of the following \begin{align} \frac{d\hat{x}}{dy}=? \end{align} I recall there is reference said that $\frac{d\hat{x}}{dy} = {\sigma^2}\text{var}[x|y]$. Is it true? How can we prove it?