Do you actually have a proof?

Yes. But it's not deep; I recommend trying yourself before consulting the answer. It follows straightforwardly from the fact that the integral of x(dx/dt) is (x^2)/2. The rest is bookkeeping to eliminate edge cases.

I didn't trouble to state the result with complete precision in the OP. For reference, here is an exact formulation (Theorem 2 of the linked note):

Let x be a differentiable real function. If the averages of x and dx/dt over the whole real line exist, and the correlation of x and dx/dt over the whole real line exists, then the correlation is zero.