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