Let $E\to M$ and $F\to M$ be vector bundles. The structure of their tangents $TE$ and $TF$ is well known. In particular, connectors map $K_E: TE \to E\times_M E$ and $K_F: TF \to F\times_M F$ induce isomorphisms $TE \simeq E \times_M (E\oplus TM)$ and $TF \simeq F \times_M (F\oplus TM)$ (I am using here a fibered product notation, rather than the equivalent pullbacks).
Consider now the vector bundle $E\otimes F\to M$. Its tangent can be characterized in the same way as the tangent bundle of every vector bundle. My question is whether there exists a canonical isomorphism of $T(E\otimes F)$ involving the tangent bundles $TE$ and $TF$. I would suspect a positive answer, which in particular looks like a "Leibniz rule". Also, what is the connector $K_{E\otimes F}$ induced by $K_E$ and $K_F$? After all, covariant derivative of tensor products satisfy a Leibniz rule.
The same question can be asked regarding the vector bundle $\operatorname{Hom}(E,F)$.