Let $f=N(\mu,\sigma^2)$ be a univariate normal distribution with mean $\mu$ and variance $\sigma^2$ and let $f_1 = N(\mu+\epsilon,\sigma^2)$ and $f_2=N(\mu,(\sigma+\epsilon)^2)$ be some small perturbations to $f$. Are there any statistical metrics $D(\cdot,\cdot)$ (e.g. Kolmogorov-Smirnov, Wasserstein, Prokhorov, etc.) for which we can say something about the derivative $$\frac{d}{d\epsilon} D(f,f_1)$$ or $$\frac{d}{d\epsilon} D(f,f_2)$$ evaluated at $\epsilon=0$? How about if $f$ were, say, an exponential distribution instead?