Polar coordinates, Gauss Lemma

Question

I don't understand the following statement from the Wikipedia page "Normal coordinates":

I don't see how this follows from the Gauss Lemma. The statement of the Gauss Lemma I know is: Let $(M,g)$ be a Riemannian manifold. Forall $p \in M$, $x \in D_p \subset T_pM$, $v,w \in T_x(T_pM)$ it holds

$g_{\exp_p(x)} (d \exp _p(v), d \exp _p(w))=g_p(v,w)$

Now $r : U \setminus \{p\} \rightarrow (0, \infty)$ correct? Then how is even $\dfrac{\partial}{\partial r}$ defined?