  If E is an elliptic curve defined over a number field and p is a prime of
good ordinary reduction for E, a theorem of Rubin relates the p-adic height
pairing on the p-power Selmer group of E to the first derivative of a
cohomologically defined p-adic L-function attached to E. Bertolini and Darmon
have defined a sequence of "derived" p-adic heights. In this paper we give an
alternative definition of the p-adic height pairing and prove a generalization
of Rubin's result, relating the derived heights to higher derivatives of p-adic
L-functions. We also relate degeneracies in the derived heights to the failure
of the Selmer group of E over a Z_p-extension to be "semi-simple" as an Iwasawa
module, generalizing results of Perrin-Riou.
