Exponential topological fields with a generic derivation

We axiomatize a class of existentially closed differential expansions of exponential topological fields where the derivation is an E-derivation. We apply our results to differential expansions of, on the one hand the field of real numbers endowed with exp(x), the classical exponential function defined by its power series expansion, and on the other hand the field of p-adic numbers endowed with the function exp(px) defined on the subring of p-adic integers where p is a prime number strictly bigger than 2 (or with exp(4x) when p=2).