Intersection of an ideal with a subset

Question

Suppose $V=\mathbb{C}[x,y]$ and consider the ideal $I=\langle x-y,\frac{W(x)-W(y)}{x-y}\rangle$, where $W$ is a polynomial in one variable. Now I want to compute the intersection $I \cap K$ where $K=\mathbb{C}[x]$. My intuition suggests that the intersection has to be something like $\langle \partial_x W \rangle$. But I don't find the right way of trying to state this, any hints are welcome