Intrinsic definition of arc length

Question

Given a Riemannian manifold $\mathcal M$ with metric $\langle \cdot, \cdot \rangle$, we define the length of a smooth curve $\gamma : I \to \mathcal M$ as \begin{equation} L(\gamma) = \int_I \sqrt{\langle \gamma'(t), \gamma'(t) \rangle}_{\gamma(t)} \ \text{d}t. \end{equation} Why are we allowed to do this, and not define the differential of a smooth map $F : \mathcal M \to \mathcal N$ by $\text{d}F_p : \gamma'(0) \mapsto (F \circ \gamma)'(0)$? I'm restricted to the non-embedded case here.