If I understand the question correctly, we are given real numbers $0 \leq x_1 \leq \dots \leq x_n$, and real numbers $0 < t_1 < \dots < t_n$. We also have some unknown differentiable function $x(t)$ that satisfies:
$$
x_i \leq x(t_i) < x_i+1 \quad for\quad i=1,2,\dots ,n
$$
Our goal is to estimate $\frac{d}{dt}x(t)$.
Suppose we are interested in the derivative at some point $t_0$. It is clear that the constraints above do not constrain the possible values of the derivative at $t_0$. Indeed, given any set of $x_i$ and $t_i$, one could find an admissible function $x(t)$ for which the derivative at $t_0$ is anything we want it to be. So unless we know something more about the "law of motion" governing x, there isn't a reasonable way to estimate the velocity.
In the absence of more information, I'll just assume that x is a polynomial. This will enforce some sort of smoothness. Finding the lowest-degree polynomial that satisfies all the constraints can be cast as a linear program. To see how, suppose that our polynomial is of degree k. Namely, $x(t) = a_0 + a_1 t + \dots + a_{k}t^k$. Then, consider the LP:
\begin{align*}
minimize &\quad z \\
\text{subject to:}&\quad 0 \leq x(t_i) \leq z\quad i=1,\dots,n
\end{align*}
Start with $n=0$, and keep increasing until you find an optimal $z$ that is less than 1. Now that you have your polynomial approximation for $x$, you can easily evaluate its derivative anywhere.