Should the input of a Kalman filter always be a signal and its derivative?
A state variable and its derivative are often included as inputs to a Kalman filter, but this is not required. The essence of the Kalman framework is that the system in question has some internal state that you are trying to estimate. You estimate those state variables based on your measurements of that system's observables over time. In many cases, you can't directly measure the state that you're interested in estimating, but if you know a relationship between your measurements and the internal state variables, you can use the Kalman framework for your problem.

There is a good example of this on the Wikipedia page. In that example, 1-dimensional linear motion of an object is considered. The object's state variables consist of its position versus time and its velocity on the one-dimensional line of movement. The example assumes that the only observable is the object's position versus time; its velocity is not observed directly. Therefore, the filter structure "infers" the velocity estimate based on the position measurements and the known relationship between velocity and position (i.e. $\dot{x_k} \approx \frac{(x_k - x_{k-1})}{\Delta t}$ if acceleration is assumed to be slowly-varying).