Normal bundle of the diagonal

Question

The normal bundle of a smooth submanifold $X \subset Y \subset R^N$ is defined as $N(X,Y)=\{(x,v) : x \in X, v \in T_xY, v \perp T_xX\}$. I want to show that $TX \cong N(\Delta,X\times X)$, where $\Delta$ is the diagonal submanifold of $X \times X$. Intuitively to me, the points of $N(\Delta,X \times X)$, should be of the form $(x,v-v)$ but I have no idea how to show that. I showed that the normal bundle is locally trivial, but I don't see if this can help in any way. Can someone help please?