Is the first derivative operation on a signal a causal system?
If the derivative exists at the given point, then it doesn't matter if you look (infinitesimally) into the future or into the past, you can do both, because both will give the same result:

$$x'(t)=\lim_{h\to 0}\frac{x(t+h)-x(t)}{h}=\lim_{h\to 0}\frac{x(t)-x(t-h)}{h}\tag{1}$$

So a differentiator can be (theoretically) implemented by a causal system.

Clearly, the system is not stable, because a bounded input can cause an unbounded output. E.g., a step at the input causes an unbounded output signal. So if by realizable we understand causal and stable, the ideal differentiator is not realizable. However, good approximations in a certain frequency range can be realized with small errors, in the continuous domain as well as in the discrete domain.