I am looking for a link between the "uncertainty propagation formula" and the Cramér-Rao lower bound (CRLB) of a function of normally distributed independent variables (not necessarily identical).
Assuming a function $f(\left\{x_i\right\})$ with the independent parameters/random variables $\left\{x_i\right\}$ having a mean and variance equal to respectively ${\mu_i}$ and ${\sigma_i}^2$, the variance of $f$, i.e. $\Delta {f}^2$ can be either obtained through the widely used "uncertainty propagation" formula: $$\Delta {f}^2 = \sum\limits_{i=1}^n \left\vert\frac{\partial f}{\partial x_i}\right\vert^2 {\sigma_i}^2$$
However, a more formal approach would be the use of the CRLB. Assuming $x_i \sim \mathscr{N}(x_i,{\sigma_i}^2)$, one can show that the expected Fisher information matrix elements can be expressed as:
$$\mathbf{I}_{ii}^f = \mathbb{E}\left(-\frac{\partial^2\log(\mathscr{L})}{\partial {x_i}^2}\right) = \frac{1}{{\sigma_i}^2}\left(\frac{\partial f}{\partial {x_i}}\right)^2$$
Is there a way to fall back on the first formula from the CRLB? Are there any extra assumptions to be made?