The exterior derivative satisfies the property
$d(f\circ g)=df\circ dg$,
which is the generalization of the chain rule (dg and df are taken at the points a and g(a) respectively). Is there a generalization of this rule to connections? In other words, is there a context in which the following chain rule makes sense
$\nabla_{f\circ g}(f\circ g)=\nabla_f f\circ \nabla_g g$
where different connections are defined for each map. Naturally the third connection would have to be related to the other two in some way. Would this composed connection be uniquely defined?