What is the partial derivative of a section?

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I made an attempt to understand the jet bundle after reading up on fiber bundles but I got stuck at the first definition, how is the partial derivative of a section defined? The total space contains fibers (homeomorphic to base point neighbourhood x model fiber) but it is not even certain that plus or multiplication is defined, it feels like something implicit is lost to me here, what restrictions are there on the total space in the definition of a jet bundle? I saw somewhere that the partials should be taken on the components of the sections, so I guess maybe the total space needs an ordered frame of sections with scalars that somehow can be differentiated? I saw mention of a smooth fiber bundle, so maybe that holds the key to what I am missing (haven't found the definition of one though).