What is Gaussian approximation for the variance of a function? In Orre 2000, the author provides an asymptotic approach to computing the variance of information component and conditioned posterior distribution.
In part 2.2 weights and information components

So far we have used the Gauss' approximation for the variance of a function, i.e.
$$V[g(X_1,\cdots,X_k)]\approx\sum_{i=1}^kV(X_i)(\partial g/\partial\mu_i)^2$$ and not included covariant terms

In part 2.3 Variance of conditioned posterior distribution

To calculate the variance of $P(a_j|D)$, below, we do a logarithmic exponential transformation, using Gaussian approximation for variance of a function
$$[V(g(X))\approx V(X)((\partial g/\partial X)(E(X)))^2]$$ thus $$V(X)=V(e^{log[X]})\approx V(log[X])E(X)^2$$

We can derive the equation directly in part 2.2 by using Taylor expansion as
$$
Var(g(X))\simeq Var[g(\mu_X)+g'(\mu_X)(X-\mu_X)]\\
=Var(X-\mu_X)[g'(\mu_X)]^2\\
=Var(X)[g'(\mu_X)]^2
$$
But how to derive the equation in part 2.3? Furthermore, I'm not sure why the author uses "Gaussian approximation" here, since "Taylor expansion" should be more accurate.

Note, that the covariance of a random variable $X$ is a matrix that does not vary with $X$. But the derivative of a function $g$ is usually not constant, so it has to be taken at a certain point, e.g. the expectation $E[X]$. Then, the approximation consists in replacing the function $g$ with its linearization at $E[X]$. And how the covariance matrix transforms with a linear map is well known. This is the first approximation you cite from part 2.3.
The second approximation you cite from 2.3 is just an application of the first equation of part 2.3 with $g(Y):= \exp(Y), Y:= \log X$:
$$
\begin{align}
Var(X) &= Var(\exp(\log(X)))\\
     &= Var(\exp Y)\\
     &\approx Var(Y) \left(\frac{d\exp(Y)}{dY}|_{Y=\log E[X]}\right)^2\\
     &= Var(Y) \exp(\log E[X])^2\\
     &= Var(\log X) E[X]^2,
\end{align}
$$
where, in the third line, we used the cited approximation, but not linearized at $Y=E[Y]$ but rather at $Y=\log E[X]$.
Of course, all the above only works if $X$ is almost surely positive, otherwise, $\log X$ is not defined.
I don't know why they call it the Gauss approximation.
