Derivative and variance of random variable

Question

Suppose a random variable $X$ is smooth in some parameter $t$, and $X(t)$ has an existing variance for all $t$. Is there any relationship between $\frac{\partial}{\partial t}\mathbb V\left[X(t)\right] \quad\text{and}\quad \mathbb V\left[\frac{\partial}{\partial t}X(t)\right]$? For the mean it is well known that, under mild conditions, $\frac{\partial}{\partial t}\mathbb E[X(t)] = \mathbb E\left[\frac{\partial}{\partial t}X(t)\right]$. Does a similar result hold for variances?