How to verify the rank-nullity theorem for the linear application $\frac{d}{dx}\in \operatorname{End}(\mathbb{K}[x]_2)$?

Question

I know that the rank-nullity theorem states that let $T: V\rightarrow W$ be a linear transformation, then:

$$ \operatorname{Rank}(T)+ \operatorname{Nullity}(T)=\dim(V)$$

But since $\frac{d}{dx}\in \operatorname{End}(\mathbb{K}[x]_2)$ is also a linear transformation, I want to verify the rank-nullity theorem with this particular situation.