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The main goal of this article is to relate the second order derivative of the Mazur-Tate-Teitelbaum $p$-adic $L$-function $L_p(E,s)$ of $E$ to Nekov\\'{a}\\v{r}'s height pairing evaluated on natural elements arising from the Beilinson-Kato elements. Along the way, we extend a Rubin-style formula of Nekov\\'a\\v{r} (or in an alternative wording, correct another Rubin-style formula of his) to apply in the presence of exceptional zeros. 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