System that has derivative of input is non causal
I am not sure why what you wrote is relevant for the causality.
Anyways, the derivative operator is depends on the future signal.
Actually this almost causal, in the sense that you need an infinitesimal environment around t to define the derivative.

Just for example, we can take 2 cases:
$$ x_1(t) = -t , x_2=|t| \Rightarrow \dot x_1(t)=-1, \dot x_2(t) =undefined$$
$$\Rightarrow y_1(t)\neq y_2(t)$$
despite $\forall t\leq 0 : x_1(t)=x_2(t)$