# Estimate variance of a function given variances of variables

I'm given the mean and the standard deviation of N Gaussian random variables $$A$$, $$B$$, $$C$$, $$D$$... I compute the function $$f:=f(A, B, C, D...) = \frac { AB... }{ CD... }$$. How can I estimate the variance of the outcome of $$f$$?

EDIT I've edited my original post to reflect a more general case.

The general issue of how errors in a function of more than one random variable are passed through in a function of those variables is called "error propagation", which is a tag on this site. This page gives a pretty thorough explanation for the ratio case. Follow the error-propagation tag on this site, or do web searches for "error propagation" and "product" for the general case (as the $C$ or $D$ terms in the denominator can be written as products of the numerator with functions $1/C$, $1/D$, etc.)
If errors in $X$ and $Y$ are approximately proportional to their values, then you can analyze the logarithm of the ratio, and the complicated product calculations become a simple additive case, $log(f)= log(X) - log(Y)$ for the original question about the ratio $X/Y$. Then the standard law for the variance of a sum/difference of random variables applies. For the more general case in the edited question, logarithms similarly simplify if appropriate.