Invertibility of an ideal differentiator
It is not. It suffices to find a counter example. Let me tell you that "I am flat". Could you derive my actual value?

So, any constant function $f(t) =c$ is differentiable, and have the same derivative, $f'(t)=0$. Only from knowing that  $f'(t)=0$, you cannot recover the original (constant) function. Hence the system is not invertible.