General correlation between function with itself and other input data

Question

• I've collected data for a function F = f(g(x)), for different function shapes g(x).
• The goal is to predict values of F for general case, i.e. for arbitrary g(x).
• It's know that function derivative dF/dx depends on history of development of dF/dx and g(x) (and maybe dg/dx); this connection is something I would like to inspect based on the collected data

In this case, which are possible techniques for finding correlations, i.e. connectivity between dF/dx and other variables?