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?



Are there any statistical metrics that satisfy this kind of condition?
According to Proposition 7 on p. 236 of [Givens and Shortt](https://projecteuclid.org/euclid.mmj/1029003026), the $L^2$ Wasserstein distance between $N(\mu\_1,\sigma\_1^2)$ and $N(\mu\_2,\sigma\_2^2)$ is the Euclidean distance between the points $(\mu\_1,\sigma\_1)$ and $(\mu\_2,\sigma\_2)$. So, the derivative of this Wasserstein distance is straightforward. 


