If an estimator concentrates around its mean, a smooth function of it looks locally like the tangent line (or plane), so the function’s variance is the input variance scaled by the squared derivative. That one move produces standard errors for log-transformed estimates, survival-curve confidence bands via Greenwood’s formula, and, in its two-variable form, the variance of ratio metrics.
The ratio case is the experimentation workhorse: metrics like clicks per session, computed at a finer grain than the randomization unit, are ratios of per-user means whose numerator and denominator co-move. The delta method prices that covariance in; the naive calculation that treats the fine-grained events as independent understates the variance and quietly inflates false-positive rates.