Tu's book "Differential Geometry" makes the claim on p79 that for a $C^{\infty}$ vector bundle $\pi : E \to M$, over a trivializing open set U any connection $\nabla_Xs$ for X a vector field on M and $s\in\Gamma(U,E)$ can be computed "by linearity and the Leibniz rule" via the connections $\nabla_Xe_i$ where $e_i$ is an element of the basis $e_1,...,e_r$ of the frame over U.
I would like to know exactly how this is done. My naive guess was that, expanding s in terms of the frame basis to get $s = a^ie_i$, since $\nabla_Xs = (Xa^i)e_i = a^i(Xe_i)$ we could then substitute in $Xe_{i} = \nabla_Xe_{i}$ to get $a^i(\nabla_Xe_{i})$. However this makes no use of the Leibniz rule and I am not sure if the substitution I'm doing is valid. If someone could provide an explicit formula for computing $\nabla_Xs$ in terms of $\nabla_Xe_i$ I would appreciate it.