Let $P \to M$ be a principal $G$-bundle with principal connection one-form $\omega$, let $V$ and $W$ be representations of $G$, and let $E=P\times_G V$ and $F=P\times_G W$ be associated vector bundles, with induced covariant derivatives $\nabla^E$ and $\nabla^F$. For smooth sections $s \in \Gamma(E)$ and $t \in \Gamma(F)$, the induced connection on $E\otimes F$ satisfies

\begin{align*} \nabla^{E\otimes F}(s\otimes t)=(\nabla^E s)\otimes t+s\otimes(\nabla^F t). \end{align*}

For every smooth section $\alpha \in \Gamma(E^*)$, the induced connection on $E^*$ satisfies

\begin{align*} d(\alpha(s))=(\nabla^{E^*}\alpha)(s)+\alpha(\nabla^E s). \end{align*}