Does Kunneth formula work for sheaf cohomology with coefficients on the tangent bundle?

Question

I want to calculate $H^1(X, TX)$, where $X=\mathbb{C} \times \mathbb{C}P^2$ and $TX$ is the sheaf of holomorphic sections of the tangent bundle of $X$. I do not know if the Kunneth formula can be used in this situation. Would there be a simple way to compute this?