Does Differentiation Cause a Phase Shift

Question

I was in class today, and a professor was explaining why capacitor and inductors cause phase shifts in voltages and currents. He mentioned that, mathematically, integration and differentiation are responsible for this. I can understand why this is the case for sine and cosine. But I would like to know if it true generally that the integral and derivative of a vector quantity will be shifted by 90 degrees from the original function.