A question about isometries

Question

Let $S$ be a regular connected surface and let $\varphi , \psi : S \to S$ two isometries and we suppose that exists $p \in S$, with $\varphi(p) = \psi(p)$, such that $d {\varphi}_p = d {\psi}_p$. I have to prove that $\varphi = \psi$ in $S$. I have tried to use Egregium Gauss theorem or Minding theorem but I haven't obtain any result. Can you help me? Thank you very much.