This is a big subject. There are many, many different notions of derivative in different contexts, all of which derive in some way from jet bundles. This question was considered in great generality in the foundational works on so-called 'differential invariants' by Lie, Cartan, and their followers and was reconsidered in modern times by many people. Some of the basic ideas in modern form can be found in R. Palais' Natural operations on differential forms and there is a more comprehensive modern treatment in the book Natural operations in differential geometry by Kolar, Michor, and Slovak.
Basically, there are methods for defining and computing the invariants of a given geometric structure under the action of some specified (pseudo-)group of transformations, and they yield, as special cases, the exterior derivative of exterior forms, the covariant derivatives of functions with respect to an affine connection, the Schwarzian derivative of a function with respect to linear fractional transformations, and a host of others. Even the Levi-Civita connection of a Riemannian metric can be regarded as the natural first derivative of the metric, which gives a simple example of a (nonlinear) derivative that is not a tensor. Using Cartan's method of equivalence, for example, one can define the invariants to all orders of most $G$-structures and test them for equivalence under various pseudo-groups.