It is a commonplace that correlation does not imply causality, however eyebrow-wagglingly suggestive it may be of causal hypotheses. It is less commonly noted that causality does not imply correlation either. It is quite possible for two variables to have zero correlation, and yet for one of them to be completely determined by the other.
That's actually an interesting issue in control systems. IIRC, if you set up a system so that some variable B is a function of the time-derivative of A, B=f( dA(t)/dt ), and it requires you to know dA(T)/dt to compute B(T), such a system is called "acausal". I believe this is because you can't know dA(T)/dt until you know A(t) after time T.
So any physically-realizable system that depends on the time-derivative of some other value, is actually depending on the time-derivative at a previous point in time.
In contrast, there is no such problem for the integral. If I only know the time series of A(t) up to time T, then I know the integral of A up to time T, and such a relationship is not acausal.
In the general case, for a relationship between two systems where B is a function of A, the transfer function from A to B, num(s)/den(s) must be such that the deg(num) <= deg(den), where deg() denotes the degree of a polynomial.
(The transfer function is ratio of B to A in the Laplace domain, usually given the variable s to replace t. Multiplying by s in the Laplace domain corresponds to differentiation in the time domain, and dividing by s is integration.)
(edit to clarify, then again to clarify some more)