The delta method, standard errors for a transformed estimator

You have an estimator $\hat\theta$ with $\operatorname{Var}(\hat\theta)$ known, but you actually care about a nonlinear function $g(\hat\theta)$, say a log-odds, a ratio, or a volatility.

Derive the approximate variance and asymptotic distribution of $g(\hat\theta)$, and apply it to the log-odds of a proportion.