The precision of a plug-in survival probability

Lifetimes $X_1, \dots, X_n$ are i.i.d. Exponential with rate $\lambda$, and $\hat\lambda = 1/\bar X$ is the maximum likelihood estimator. You want the survival probability past a fixed time $t$, $S = P(X > t) = e^{-\lambda t}$, estimated by the plug-in $\hat S = e^{-\hat\lambda t}$.

Use the delta method to find the approximate variance of $\hat S$.